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Full text of "The scientific papers of James Clerk Maxwell"

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THE SCIENTIFIC PAPERS OF 

JAMES CLERK MAXWELL 



Edited by W. D. NIVEN, M.A., F.R.S. 



Two Volumes Bound As One 



DOVER PUBLICATIONS, INC., NEW YORK 



All rights reserved under Pan American and In- 
ternational Copyright Conventions. 



Published in Canada by General Publishing Com- 
pany, Ltd., 30 Lesmill Road, Don Mills, Toronto, 
Ontario. 

Published in the United Kingdom by Constable 
and Company, Ltd., 10 Orange Street, London 
W. C. 2. 



This Dover edition, first published in 1965, is an 
unabridged and unaltered republication of the work 
first pubhshed by Cambridge University Press in 
1890. This edition is published by special arrange- 
ment with Cambridge University Press. 

The work was originally pubhshed in two separate 
volumes, but is now published in two volumes 
bound as one. 



Library of Congress Catalog Card Number: A53 -9813 



Manufactured in the United States of America 

Dover Publications, Inc. 

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THE SCIENTIFIC PAPERS OF 

JAMES CLERK MAXWELL 



Edited by W. D. NIVEN, M.A., F.R.S, 



Volume One 



TO HIS GRACE 

THE DUKE OF DEVONSHIRE K.G. 

CHANCELLOR OF THE UNIVERSITY OF CAMBRIDGE 

FOUNDER OF THE CAVENDISH LABORATORY 

THIS MEMORIAL EDITION 

OF 

THE SCIENTIFIC PAPERS 

OF 

THE FIRST CAVENDISH PROFESSOR OF EXPERIMENTAL PHYSICS 

IS 

BY HIS GRACE'S PERMISSION 

RESPECTFULLY AND GRATEFULLY DEDICATED 



SHORTLY after the death of Professor James Clerk Maxwell a Committee was 
formed, consisting of graduate members of the University of Cambridge and 
of other friends and admirers, for the purpose of securing a fitting memorial of 
him. 

The Committee had in view two objects : to obtain a likeness of Professor 
Clerk Maxwell, which should be placed in some public building of the Uni- 
versity ; and to collect and publish his scattered scientific writings, copies of 
which, so far as the funds at the disposal of the Committee would allow, 
should be presented to learned Societies and Libraries at home and abroad. 

It was decided that the likeness should take the form of a marble bust. 
This was executed by Sir J. E. Boehm, R.A., and is now placed in the 
apparatus room of the Cavendish Laboratory. 

In carrying out the second part of their programme the Committee 
obtained the cordial assistance of the Syndics of the University Press, who 
willingly consented to publish the present work. At the request of the Syndics, 
Mr W. D. Niven, M.A., Fellow and Assistant Tutor of Trinity College and 
now Director of Studies at the Royal Naval College, Greenwich, undertook the 
duties of Editor. 

The Committee and the Syndics desire to take this opportunity of 
acknowledging their obligation to Messrs Adam and Charles Black, Publishers 
of the ninth Edition of the EiicyclopcEdia Biitannica, to Messrs Taylor and 
Francis, Publishers of the London, Edinburgh, and Dublin Philosophical Maga- 
zine and Journal of Science, to Messrs Macmillan and Co., Publishers of 
Nature and of the Cambridge and Dublin Mathematical Joui-nal, to Messrs 
Metcalfe and Co., Publishers of the Quarterly Journal of Pure and Applied 
Mathematics, and to the Lords of the Conmiittee of Council on Education, 
Proprietors of the Handbooks of the South Kensington Museum, for their 
courteous consent to allow the articles which Clerk Maxwell had contributed to 
these publications to be included in the present work ; to Mr Norman Lockyer 
for the assistance which he rendered in the selection of the articles re-printed 
from Nature; and their further obligation to Messrs Macmillan and Co. for 
permission to use in this work the steel engravings of Faraday, Clerk Maxwell, 
and Helmholtz from the Nature Series of Portraits. 



Numerous and important Papers, contributed by Clerk Maxwell to the 
Transactions or Proceedings of the Royal Societies of London and of Edinburgh, 
of the Cambridge Philosophical Society, of the Royal Scottish Society of Arts, 
and of the London Mathematical Society; Lectures delivered by Clerk Maxwell 
at the Royal Institution of Great Britain pubHshed in its Proceedings; as well 
as Communications and Addresses to the British Association published in its 
Reports, are also included in the present work with the sanction of the above 
mentioned learned bodies. 

The Essay which gained the Adams Prize for the year 1856 in the 
University of Cambridge, the introductory Lecture on the Study of Experimental 
Physics delivered in the Cavendish Laboratory, and the Rede Lecture delivered 
before the University in 1878, complete this collection of Clerk Maxwell's scientific 
writings. 

The diagrams in this work have been re-produced by a photographic 
process from the original diagrams in Clerk Maxwell's Papers by the Cambridge 
Scientific Instrument Company. 

It only remams to add that the footnotes inserted by the Editor are 
enclosed between square brackets. 

Cambridge, Augv^t, 1890. 



PEEFACE. 



CLERK MAXWELL'S biography has been written by Professors Lewis Campbell and 
Wm. Garnett with so much skill and appreciation of their subject that nothing further 
remains to be told. It would therefore be presumption on the part of the editor of his 
papers to attempt any lengthened narrative of a biographical character. At the same time 
a memorial edition of an author's collected writings would hardly be complete without 
some account however slight of his life and works. Accordingly the principal events of 
Clerk Maxwell's career will be recounted in the following brief sketch, and the reader 
who wishes to obtain further and more detailed information or to study his character in 
its social relations may consult the interesting work to which reference has been made. 

James Clerk Maxwell was descended from the Clerks of Penicuick in Midlothian, 
a well-known Scottish family whose history can be traced back to the IGth century. The 
first baronet served in the parliament of Scotland. His eldest son, a man of learning, 
was a Baron of the Exchequer in Scotland. In later times John Clerk of Eldin a 
member of the family claimed the credit of having invented a new method of breaking 
the enemy's line in naval warfare, an invention said to have been adopted by Lord 
Rodney in the battle which he gained over the French in 1782. Another John Clerk, 
son of the naval tactitian, was a lawyer of much acumen and became a Lord of the 
Court of Session. He was distinguished among his Edinburgh contemporaries by his ready 
and sarcastic wit. 

The father of the subject of this memoir was John, brother to Sir George Clerk of 
Penicuick. He adopted the surname of Maxwell on succeeding to an estate in Kirkcud- 
brightshire which came into the Clerk family through marriage with a Miss Maxwell. It 
cannot be said that he was possessed of the energy and activity of mind which lead 
to distinction. He was in truth a somewhat easy-going but shrewd and intelligent 
man, whose most notable characteristics were his perfect sincerity and extreme benevolence. 
He took an enlightened interest in mechanical and scientific pursuits and was of an 
essentially practical turn of mind. On leaving the University he had devoted himself 
to law and was called to the Scottish Bar. It does not appear however that he met 
mth any great success in that profession. At all events, a quiet life in the country 



X PREFACE. 

presented so many attractions to his wife as well as to himself that he was easily induced 
to relinquish his prospects at the bar. He had been married to Frances, daughter of 
Robert Cay of N. Charlton, Northumberland, a lady of strong good sense and resolute 
character. 

The country house which was their home after they left Edinburgh was designed 
by John Clerk Maxwell himself and was built on his estate. The house, which was named 
Glenlair, was surrounded by fine scenery, of which the water of Urr with its rocky and 
wooded banks formed the principal charm. 

James was bom at Edinburgh on the 13th of June, 1831, but it was at Glenlair 
that the greater part of his childhood was passed. In that pleasant spot under healthful 
influences of all kinds the child developed into a hardy and ccirageous boy. Not 
precociously clever at books he was yet not without some signs of future intellectual 
strength, being remarkable for a spirit of inquiry into the caupjs and connections of the 
phenomena around him. It was remembered afterwards when he had become distinguished, 
that the questions he put as a child shewed an amount of thoughtfulness which for his 
years was very unusual. 

At the age of ten, James, who had lost his mother, was placed under the charge of 
relatives in Edinburgh that he might attend the Edinburgh Academy. A charming account 
of his school days is given in the narrative of Professor Campbell who was Maxwell's 
schoolfellow and in after life an intimate friend and constant correspondent. The child is 
father to the man, and those who were privileged to know the man Maxwell will easily 
recognise Mr Campbell's picture of the boy on his first appearance at school, — the home- 
made garments more serviceable than fashionable, the rustic speech and curiously quaint 
but often humorous manner of conveying his meaning, his bewilderment on first undergoing 
the routine of schoolwork, and his Spartan conduct under various trials at the hands of 
his schoolfellows. They will further feel how accurate is the sketch of the boy become 
accustomed to his surroundings and rapidly assuming the place at school to which his 
mental powers entitled him, while his superfluous energy finds vent privately in carrying 
out mechanical contrivances and geometrical constructions, in reading and even trying his 
hand at composing ballads, and in sending to his father letters richly embellished with 
grotesquely elaborate borders and drawings. 

An event of his school-days, worth recording, was his invention of a mechanical method 
of drawing certain classes of Ovals. An account of this method was printed in the 
Proceedings of the Royal Society of Edinburgh and forms the first of his writings 
collected in the present work. The subject was introduced to the notice of the Society 
by the celebrated Professor James Forbes, who from the first took the greatest possible 
interest in Maxwell's progress. Professor Tait, another schoolfellow, mentions that at the 
time when the paper on the Ovals was written. Maxwell had received no instruction in 
Mathematics beyond a little Euclid and Algebra. 



PREFACE. aa 

In 1847 Maxwell entered the University of Edinburgh where he remained for three 
sessions. He attended the lectures of Kelland in Mathematics, Forbes in Natural Philosophy, 
Gregory in Chemistry, Sir W. Hamilton in Mental Philosophy, Wilson (Christopher North) 
in Moral Philosophy. The lectures of Sir W. Hamilton made a strong impression upon 
him, in stimulating the love of speculation to which his mind was prone, but, as might 
have been expected, it was the Professor of Natural Philosophy who obtained the chief share 
of his devotion. The enthusiasm which so distinguished a man as Forbes naturally inspired 
in young and ardent disciples, evoked a feeling of personal attachment, and the Professor, on 
his part, took special interest in his pupil and gave to him the altogether unusual 
privilege of working with his fine apparatus. 

What was the nature of this experimental work we may conjecture from a perusal of 
his paper on Elastic Solids, written at that time, in which he describes some experiments 
made with the view of verifying the deductions of his theory in its application to Optics. 
Maxwell would seem to have been led to the study of this subject by the following cir- 
cumstance. He was taken by his uncle John Cay to see William Nicol, the inventor of 
the polarising prism which bears his name, and was shewn by Nicol the colours of unan- 
nealed glass in the polariscope. This incited Maxwell to study the laws of polarised light 
and to construct a rough polariscope in which the polariser and analyser were simple glass 
reflectors. By means of this instrument he was able to obtain the colour bands of unannealed 
glass. These he copied on paper in water colours and sent to Nicol. It is gratifpng to 
find that this spirited attempt at experimenting on the part of a mere boy was duly 
appreciated by Nicol, who at once encouraged and delighted him by a present of a couple of 
his prisms. 

The paper alluded to, viz. that entitled "On the Equilibrium of Elastic Solids," was 
read to the Royal Society of Edinburgh in 1850. It forms the third paper which Maxwell 
addressed to that Society. The first in 1846 on Ovals has been abready mentioned. The 
second, under the title "The Theory of Rolling Curves," was presented by Kelland in 1849. 

It is obvious that a youth of nineteen years who had been capable of these efforts 
must have been gifted with rare originality and with great power of sustained exertion. 
But his singular self-concentration led him into habits of solitude and seclusion, the tendency 
of which was to confirm his peculiarities of speech and of manner. He was shy and 
reserved with strangers, and his utterances were often obscure both in substance and in 
his manner of expressing himself, so many remote and unexpected allusions perpetually 
obtruding themselves. Though really most sociable and even fond of society he was 
essentially reticent and reserved. Mr Campbell thinks it is to be regretted that Maxwell 
did not begin his Cambridge career eai'lier for the sake of the social intercourse which 
he would have found it difficult to avoid there. It is a question, however, whether in 
losing the opportunity of using Professor Forbes' apparatus he would not thereby have lost 
what was perhaps the most valuable part of his early scientific training. 



XU PREFACE. 

It was originally intended that Maxwell should follow his father's profession of advocate, 
but this intention was abandoned as soon as it became obvious that his tastes lay in a 
direction so decidedly scientific. It was at length determined to send him to Cambridge 
and accordingly in October, 1850, he commenced residence in Peterhouse, where however he 
resided during the Michaelmas Term only. On December 14 of the same year he migrated 
to Trinity College. 

It may readily be supposed that his preparatory training for the Cambridge course 
was far removed from the ordinary type. There had indeed for some time been practically 
no restraint upon his plan of study and his mind had been allowed to follow its natural 
bent towards science, though not to an extent so absorbing as to withdraw him from 
other pursuits. Though he was not a sportsman, — indeed sport so called was always repugnant 
to him — he was yet exceedingly fond of a country life. He was a good horseman and a 
good swimmer. Whence however he derived his chief enjoyment may be gathered from the 
account which Mr Campbell gives of the zest with which he quoted on one occasion the 
lines of Bums which describe the poet finding inspiration while wandering along the banks 
of a stream in the free indulgence of his fancies. Maxwell was not only a lover of poetry 
but himself a poet, as the fine pieces gathered together by Mr Campbell abundantly testify. 
He saw however that his true calling was Science and never regarded these poetical 
efforts as other than mere pastime. Devotion to science, already stimulated by successful 
endeavour, a tendency to ponder over philosophical problems and an attachment to English 
literature, particularly to English poetry, — these tastes, implanted in a mind of singular 
strength and purity, may be said to have been the endowments with which young Maxwell 
began his Cambridge career. Besides this, his scientific reading, as we may gather from his 
papers to the Royal Society of Edinburgh referred to above, was already extensive and 
varied. He brought with him, says Professor Tait, a mass of knowledge which was really 
immense for so young a man but in a state of disorder appalling to his methodical 
private tutor. 

Maxwell's undergraduate career was not marked by any specially notable feature. His 
private speculations had in some measure to be laid aside in favour of more systematic 
study. Yet his mind was steadily ripening for the work of his later years. Among those 
with whom he was brought into daily contact by his position, as a Scholar of Trinity 
College, were some of the brightest and most cultivated young men in the University. In 
the genial fellowship of the Scholars' table Maxwell's kindly humour found ready play, while 
in the more select coterie of the Apostle Club, formed for mutual cultivation, he found a field 
for the exercise of his love of speculation in essays on subjects beyond the lines of the 
ordinary University course. The composition of these essays doubtless laid the foundation 
of that literary finish which is one of the characteristics of Maxwell's scientific writings. 
His biographers have preserved several extracts on a variety of subjects chiefly of a specu- 
lative character. They are remarkable mainly for the weight of thought contained in them 
but occasionally also for smart epigrams and for a vein of dry and sarcastic humour. 



PREFACE. 



These glimpses into Maxwell's character may prepare us to believe that, with all his 
shyness, he was not without confidence in his own powers, as also appears from the account 
which was given by the late Master of Trinity College, Dr Thompson, who was Tutor when 
Maxwell personally applied to him for permission to migrate to that College. He appeared 
to be a shy and diffident youth, but presently surprised Dr Thompson by producing a 
bundle of papers, doubtless copies of those we have already mentioned, remarking " Perhaps 
these may shew you that I am not unfit to enter at your College." 

He became a pupil of the celebrated William Hopkins of Peterhouse, under whom his 
course of study became more systematic. One striking characteristic was remarked by his 
contemporaries. Whenever the subject admitted of it he had recourse to diagrams, though 
his fellow students might solve the question more easily by a train of analysis. Many 
illustrations of this manner of proceeding might be taken from his writings, but in 
truth it was only one phase of his mental attitude towards scientific questions, which 
led him to proceed from one distinct idea to another instead of trusting to symbols and 
equations. 

Maxwell's published contributions to Mathematical Science during his undergraduate career 
were few and of no great importance. He found time however to carry his investigations 
into regions outside the prescribed Cambridge course. At the lectures of Professor Stokes* 
he was regular in his attendance. Indeed it appears from the paper on Elastic Solids, 
mentioned above, that he was acquainted with some of the writings of Stokes before he 
entered Cambridge. Before 1850, Stokes had published some of his most important contri- 
butions to Hydromechanics and Optics ; and Sir W. Thomson, who was nine years' Maxwell's 
senior in University standing, had, among other remarkable investigations, called special 
attention to the mathematical analogy between Heat-conduction and Statical Electricity. 
There is no doubt that these authors as well as Faraday, of whose experimental researches 
he had made a careful study, exercised a powerful directive influence on his mind. 

In January, 1854, Maxwell's undergraduate career closed. He was second wrangler, but 
shared with Dr Routh, who was senior wrangler, the honours of the First Smith's Prize. 
In due course he was elected Fellow of Trinity and placed on the staff of College Lecturers. 

No sooner was he released from the restraints imposed by the Trinity Fellowship 
Examination than he plunged headlong into original work. There were several questions 
he was anxious to deal with, and first of all he completed an investigation on the Trans- 
formation of Surfaces by Bending, a purely geometrical problem. This memoir he presentel 
to the Cambridge Philosophical Society in the following March. At this period he also 
set about an enquiry into the quantitative measurement of mixtures of colours and the 
causes of colour-blindness. During his undergraduateship he had, as we have seen, found 
time for the study of Electricity. This had already borne fruit and now resulted in the 
first of his important memoirs on that subject,— the memoir on Faraday's Lines of Force. 
• Now Sir George Gabriel Stokes, Bart., M.P. for the University. 



Xiv PREFACE. 

The number and importance of his papers, published in 1855—6, bear witness to his 
assiduity during this period. With these labours, and in the preparation of his College 
lectures, on which he entered with much enthusiasm, his mind was fully occupied and the 
work was congenial. He had formed a number of valued friendships, and he had a variety of 
interests, scientific and literary, attaching him to the University. Nevertheless, when the chair 
of Natural Philosophy in Marischal College, Aberdeen, fell vacant, Maxwell became a candidate. 
This step was probably taken in deference to his father's wishes, as the long summer 
vacation of the Scottish College would enable him to reside with his father at Glenlair for 
half the year continuously. He obtained the professorship, but unhappily the kind intentions 
which prompted him to apply for it were frustrated by the death of his father, which took 
place in April, 1856. 

It is doubtful whether the change from the Trinity lectureship to the Aberdeen 
professorship was altogether prudent. The advantages were the possession of a laboratory and 
the long uninterrupted summer vacation. But the labour of drilling classes composed chiefly 
of comparatively young and untrained lads, in the elements of mechanics and physics, was 
not the work for which Maxwell was specially fitted. On the other hand, in a large college 
like Trinity there could not fail to have been among its undergraduate members, some of the 
most promising young mathematicians of the University, capable of appreciating his original 
genius and immense knowledge, by instructing whom he would himself have derived ad- 
vantage. 

In 1856 Maxwell entered upon his duties as Professor of Natural Philosophy at Marischal 
College, and two years afterwards he married Katharine Mary Dewar, daughter of the 
Principal of the College. He in consequence ceased to be a Fellow of Tiinity College, 
but was afterwards elected an honorary Fellow, at the same time as Professor Cayley. 

During the yeai*s 1856 — 60 he was still actively employed upon the subject of colour 
sensation, to which he contributed a new method of measurement in the ingenious instru- 
ment known as the colour-box. The most serious demands upon his powers and upon his 
time were made by his investigations on the Stability of Saturn's Rings. This was the 
subject chosen by the Examiners for the Adams Prize Essay to be adjudged in 1857, and 
was advertised in the following terms: — 

"The Problem may be treated on the supposition that the system of Rings is 
exactly or very approximately concentric with Saturn and symmetrically disposed about 
the plane of his equator and different hypotheses may be made respecting the physical 
constitution of the Rings. It may be supposed (1) that they are rigid; (2) that they 
are fluid and in part aeriform ; (3) that they consist of masses of matter not materially 
coherent. The question will be considered to be answered by ascertaining on these 
hypotheses severally whether the conditions of mechanical stability are satisfied by the 
mutual attractions and motions of the Planet and the Rings." 



PREFACE. XV 

"It is desirable that an attempt should also be made to determine on which of 

the above hypotheses the appearances both of the bright rings and the recently 

discovered dark ring may be most satisfactorily explained; and to indicate any causes 

to which a change of form such as is supposed from a comparison of modem with the 

earlier observations to have taken place, may be attributed." 

It is sufficient to mention here that Maxwell bestowed an immense amount of labour 

in working out the theory as proposed, and that he arrived at the conclusion that "the 

only system of rings which can exist is one composed of an indefinite number of unconnected 

particles revolving round the planet with different velocities according to their respective 

distances. These particles may be arranged in a series of narrow rings, or they may move 

about through each other irregularly. In the first case the destruction of the system will be 

very slow, in the second case it will be more rapid, but there may be a tendency towards 

an aiTangement in narrow rings which may retard the process." 

Part of the work, dealing with the oscillatory waves set up in a ring of satellites, 
was illustrated by an ingenious mechanical contrivance which was greatly admired when 
exhibited before the Royal Society of Edinburgh. 

This essay, besides securing the prize, obtained for its author great credit among 
scientific men. It was characterized by Sir George Airy as one of the most remarkable 
applications of Mathematics to Physics that he had ever seen. 

The suggestion has been made that it was the irregular motions of the particles which 
compose the Rings of Saturn resulting on the whole in apparent regularity and uni- 
formity, which led Maxwell to the investigation of the Kinetic Theory of Gases, his first 
contribution to which was read to the British Association in 1859. This is not unlikely, 
but it must also be borne in mind that Bernoulli's Theory had recently been revived by 
Herapath, Joule and Clausius whose writings may have drawn Maxwell's attention to the 
subject. 

In 1860 King's College and Marischal College were joined together as one institution, 
now known as the University of Aberdeen. The new chair of Natural Philosophy thus 
created was filled up by the appointment of David Thomson, formerly Professor at King's 
College and Maxwell's senior. Professor Thomson, though not comparable to Maxwell as a 
physicist, was nevertheless a remarkable man. He was distinguished by singular force of 
character and great administrative faculty and he had been prominent in bringing about 
the fusion of the Colleges. He was also an admirable lecturer and teacher and had done 
much to raise the standard of scientific education in the north of Scotland. Thus the choice 
made by the Commissioners, though almost inevitable, had the effect of making it appear 
that Maxwell failed as a teacher. There seems however to be no evidence to support such 
an inference. On the contrary, if we may judge from the number of voluntary students 
attending his classes in his last College session, he would seem to have been as popular as a 
professor as he was personally estimable. 



XVI PREFACE. 

This is also borne out by the fact that he was soon afterwards elected Professor of 
Natural Philosophy and Astronomy in King's College, London. The new appointment had 
the advantage of bringing him much more into contact with men in his own department 
of science, especially with Faraday, with whose electrical work his own was so intimately 
connected. In 1862 — 63 he took a prominent part in the experiments organised by a 
Committee of the British Association for the determination of electrical resistance in 
absolute measure and for placing electrical measurements on a satisfactory basis. In the 
experiments which were conducted in the laboratory of King's College upon a plan due 
to Sir W. Thomson, two long series of measurements were taken in successive years. In 
the first year, the working members were Maxwell, Balfour Stewart and Fleeming Jenkin ; in 
the second, Charles Hockin took the place of Balfour Stewart. The work of this Committee 
was communicated in the form of reports to the British Association and was afterwards 
republished in one volume by Fleeming Jenkin. 

Maxwell was a professor in King's College from 1860 to 1865, and this period of his 
life is distinguished by the production of his most important papers. The second memoir 
on Colours made its appearance in 1860. In the same year his first papers on the Kinetic 
Theory of Gases were published. In 1861 came his papers on Physical Lines of Force 
and in 1864 his greatest memoii' on Electricity, — a Dynamical Theory of the Electro- 
magnetic Field. He must have been occupied with the Dynamical Theory of Gases in 1865, 
as two important papers appeared in the following year, first the Bakerian lecture on the 
Viscosity of Gases, and next the memoir on the Dynamical Theory of Gases. 

The mental strain involved in the production of so much valuable work, combined 
with the duties of his professorship which required his attention during nine months of 
the year, seems to have influenced him in a resolution which in 1865 he at length 
adopted of resigning his chair and retiring to his country seat. Shortly after this he had 
a severe illness. On his recovery he continued his work on the Dynamical Theory of 
Gases, to which reference has just been made. For the next few years he led a quiet 
and secluded life at Glenlair, varied by annual visits to London, attendances at the British 
Association meetings and by a tour in Italy in 1867. He was also Moderator or Examiner 
in the Mathematical Tripos at Cambridge on several occasions, ofiBces which entailed a few 
weeks' residence at the University in winter. His chief employment during those years 
was the prepai-ation of his now celebrated treatise on Electricity and Magnetism which, 
however, was not published till 1873. He also wrote a treatise on Heat which was 
published in 1871. 

In 1871 Maxwell was, with some reluctance, induced to quit his retreat in the 
country and to enter upon a new career. The University of Cambridge had recently 
resolved to found a professorship of physical science, especially for the cultivation and 
teaching of the subjects of Heat, Electricity and Magnetism. In furtherance of this 
object her Chancellor, the Duke of Devonshire, had most generously undertaken to build 
a laboratory and furnish it with the necessary apparatus. Maxwell was invited to fill the 



PREFACE. XVU 

new chair thus formed and to superintend the erection of the laboratory. In October, 
1871, he delivered his inaugural lecture. 

The Cavendish Laboratory, so called after its founder, the present venerable chief of 
the family which produced the great physicist of the same name, was not completed 
for practical work until 1874. In June of that year it was formally presented to the 
University by the Chancellor. The building itself and the fittings of the several rooms 
were admirably contrived mainly by Maxwell himself, but the stock of apparatus was 
smaller than accorded with the generous intentions of the Chancellor. This defect must 
be attributed to the anxiety of the Professor to procure only instruments by the best 
makers and with such improvements as he could himself suggest. Such a defect therefore 
required time for its removal and afterwards in great measure disappeared, apparatus being 
constantly added to the stock as occasion demanded. 

One of the chief tasks which Maxwell undertook was that of superintending and 
directing the energies of such young Bachelors of Arts as became his pupils after 
having acquired good positions in the University examinations. Several pupils, who have 
since acquired distinction, carried out valuable experiments under the guidance of the 
Professor. It must be admitted, however, that the numbers were at first small, but perhaps 
this was only to be expected from the traditions of so many years. The Professor was 
singularly kind and helpful to these pupils. He would hold long conversations with them, 
opening up to them the stores of his mind, giving them hints as to what they might try 
and what avoid, and was always ready with some ingenious remedy for the experimental 
troubles which beset them. These conversations, always delightful and instructive, were, 
according to the account of one of his pupils, a liberal education in themselves, and were 
repaid in the minds of the pupils by a grateful affection rarely accorded to any teacher. 

Besides discharging the duties of his chair, Maxwell took an active part in conducting 
the general business of the University and more particularly in regulating the courses of 
study in Mathematics and Physics. 

For some years previous to 1866 when Maxwell returned to Cambridge as Moderator 
in the Mathematical Tripos, the studies in the University had lost touch with the great 
scientific movements going on outside her walls. It was said that some of the subjects most 
in vogue had but little interest for the present generation, and loud complaints began to 
be heard that while such branches of knowledge as Heat, Electricity and Magnetism, were 
left out of the Tripos examination, the candidates were wasting their time and energy 
upon mathematical trifles barren of scientific interest and of practical results. Into the 
movement for reform Maxwell entered warmly. By his questions in 1866 and subsequent 
years he infused new life into the examination ; he took an active part in drafting the 
new scheme introduced in 1873 ; but most of all by his writings he exerted a powerful 
influence on the younger members of the University, and was largely instrumental in 
bringing about the change which has been now effected. 



XVIU PREFACE. 

In the first few years at Cambridge Maxwell was busy in giving the final touches 
to his great work on Electricity and Magnetism and in passing it through the press. 
This work was published in 1873, and it seems to have occupied the most of his attention 
for the two previous years, as the few papers published by him during that period relate 
chiefly to subjects forming part of the contents. After this publication his contributions to 
scientific journals became more numerous, those on the Dynamical Theory of Gases being 
perhaps the most important. He also wrote a great many short articles and reviews 
which made their appearance in Nature and the Encyclopcedia Britannica. Some of these 
essays are charming expositions of scientific subjects, some are general criticisms of the 
works of contemporary writers and others are brief and appreciative biographies of fellow 
workers in the same fields of research. 

An undertaking in which he was long engaged and which, though it proved exceedingly 
interesting, entailed much labour, was the editing of the "Electrical Researches" of the Hon. 
Henry Cavendish. This work, published in 1879, has had the eflfect of increasing the 
reputation of Cavendish, disclosing as it does the unsuspected advances which that acute 
physicist had made in the Theory of Electricity, especially in the measurement of electrical 
quantities. The work is enriched by a variety of valuable notes in which Cavendish's 
views and results are examined by the light of modern theory and methods. Especially 
valuable are the methods applied to the determination of the electrical capacities of con- 
ductors and condensers, a subject in which Cavendish himself shewed considerable skill 
both of a mathematical and experimental character. 

The importance of the task undertaken by Maxwell in connection with Cavendish's 
papers will be understood from the following extract from his introduction to them. 

"It is somewhat difficult to account for the fact that though Cavendish had 
prepared a complete description of his experiments on the charges of bodies, and had 
even taken the trouble to write out a fair copy, and though all this seems to have 
been done before 1774 and he continued to make experiments in Electricity till 1781 
and lived on till 1810, he kept his manuscript by him and never published it." 

"Cavendish cared more for investigation than for publication. He would under- 
take the most laborious researches in order to clear up a difficulty which no one 
but himself could appreciate or was even aware of, and we cannot doubt that the 
result of his enquiries, when successful, gave him a certain degree of satisfaction. 
But it did not excite in him that desire to communicate the discovery to others 
which in the case of ordinary men of science, generally ensures the publication of 
their results. How completely these researches of Cavendish remained unknown to 
other men of science is shewn by the external history of electricity." 

It will probably be thought a matter of some difficulty to place oneself in the 
position of a physicist of a century ago and to ascertain the exact bearing of his 
experiments. But Maxwell entered upon this undertaking with the utmost enthusiasm and 



PREFACE. XIX 

succeeded in completely identifying himself with Cavendish's methods. He shewed that 
Cavendish had really anticipated several of the discoveries in electrical science which have been 
made since his time. Cavendish was the first to form the conception of and to measure 
Electrostatic Capacity and Specific Inductive Capacity; he also anticipated Ohm's law. 

The Cavendish papers were no sooner disposed of than Maxwell set about preparing 
a new edition of his work on Electricity and Magnetism; but unhappily in the summer 
term of 1879 his health gave way. Hopes were however entertained that when he returned 
to the bracing air of his country home he would soon recover. But he lingered through 
the summer months with no signs of improvement and his spirits gradually sank He was 
finally informed by his old fellow-student, Professor Sanders, that he could not live more 
than a few weeks. As a last resort he was brought back to Cambridge in October that he 
might be under the charge of his favourite physician, Dr Paget*. Nothing however could 
be done for • his malady, and, after a painful illness, he died on the 5th of November, 1879, 
in his 49th year. 

Maxwell was thus cut oflf in the prime of his powers, and at a time when the depart- 
ments of science, which he had contributed so much to develop, were being every day 
extended by fresh discoveries. His death was deplored as an irreparable loss to science and 
to the University, in which his amiable disposition was as universally esteemed as his genius 
was admired. 

It is not intended in this preface to enter at length into a discussion of the relation 
which Maxwell's work bears historically to that of his predecessors, or to attempt to estimate 
the effect which it has had on the scientific thought of the present day. In some of his 
papers he has given more than usually copious references to the works of those by whom 
he had been influenced; and in his later papers, especially those of a more popular nature 
which appeared in the Encyclopoedia Britannica, he has given full historical outlines of some 
of the most prominent fields in which he laboured. Nor does it appear to the present 
editor that the time has yet arrived when the quickening influence of Maxwell's mind on 
modem scientific thought can be duly estimated. He therefore proposes to himself the duty 
of recalling briefly, according to subjects, the most important speculations in which Maxwell 
engaged. 

His works have been arranged as far as possible in chronological order but they fall 
naturally under a few leading heads; and perhaps we shall not be far wrong if we place 
first in importance his work in Electricity. 

His first paper on this subject bearing the title "On Faraday's Lines of Force" was 
read before the Cambridge Philosophical Society on Dec. 11th, 1855. He had been previously 
attracted by Faraday's method of expressing electrical laws, and he here set before himself 
the task of shewing that the ideas which had guided Faraday's researches were not incon- 
sistent with the mathematical formulae in which Poisson and others had cast the laws of 
♦ Now Sir George Edward Paget, K.C.B. 



PREFACE. 



Electricity. His object, he says, is to find a physical analogy which shall help the mind 
to grasp the results of previous investigations "without being committed to any theory 
founded on the physical science from which that conception is borrowed, so that it is neither 
draw aside from the subject in the pursuit of analytical subtleties nor carried beyond the 
truth by a favorite hypothesis." 

The laws of electricity are therefore compared with the properties of an incompressible 
fluid the motion of which is retarded by a force proportional to the velocity, and the fluid 
is supposed to possess no inertia. He shews the analogy which the lines of flow of such 
a fluid would have with the lines of force, and deduces not merely the laws of Statical 
Electricity in a single medium but also a method of representing what takes place when the 
action passes from one dielectric into another. 

In the latter part of the paper he proceeds to consider the phenomena of Electro- 
magnetism and shews how the laws discovered by Ampere lead to conclusions identical with 
those of Faraday. In this paper three expressions are introduced which he identifies with 
the components of Faraday's electrotonic state, though the author admits that he has not 
been able to frame a physical theory which would give a clear mental picture of the 
various connections expressed by the equations. 

Altogether this paper is most important for the light which it throws on the principles 
which guided Maxwell at the outset of his electrical work. The idea of the electrotonic 
state had afready taken a firm hold of his mind though as yet he had formed no physical 
explanation of it. In the paper "On Physical Lines of Force" printed in the Philosophical 
Magazine, Vol. xxi. he resumes his speculations. He explains that in his former paper he 
had found the geometrical significance of the Electrotonic state but that he now proposes 
"to examine magnetic phenomena from a mechanical point of view." Accordingly he propounds 
his remarkable speculation as to the magnetic field being occupied by molecular vortices, 
the axes of which coincide with the lines of force. The cells within which these vortices 
rotate are supposed to be separated by layers of particles which serve the double purpose 
of transmitting motion from one cell to another and by their own motions constituting an 
electric current. This theory, the parent of several working models which have been devised 
to represent the motions of the dielectric, is remarkable for the detail vnth which it is 
worked out and made to explain the various laws not only of magnetic and electromagnetic 
action, but also the various forms of electrostatic action. As Maxwell subsequently gave a 
more general theory of the Electromagnetic Field, it may be inferred that he did not desire 
it to be supposed that he adhered to the views set forth in this paper in every particular; 
but there is no doubt that in some of its main features, especially the existence of 
rotation round the lines of magnetic force, it expressed his permanent convictions. In his 
treatise on "Electricity and Magnetism," Vol. ii. p. 416, (2nd edition 427) after quoting from 
Sir W. Thomson on the explanation of the magnetic rotation of the plane of the polarisation 
of light, he goes on to say of the present paper, 



PREFACE. XXI 

"A theory of molecular vortices which T worked out at considerable length was 
published in the Phil. Mag. for March, April and May, 1861, Jan. and Feb. 1862." 

- " I think we have good evidence for the opinion that some phenomenon of rotation 
is going on in the magnetic field, that this rotation is performed by a great number 
of very small portions of matter, each rotating on its own axis, that axis being parallel 
to the direction of the magnetic force, and that the rotations of these various vortices 
are made to depend on one another by means of some mechanism between them." 

"The attempt which I then made to imagine a working model of this mechanism 

must be taken for no more than it really is, a demonstration that mechanism may 

be imagined capable of producing a connection mechanically equivalent to the actual 

connection of the parts of the Electromagnetic Field." 

This paper is also important as containing the first hint of the Electromagnetic Theory 

of Light which was to be more fully developed afterwards in his third great memoir 

" On the Dynamical Theory of the Electromagnetic Field." This memoir, which was presented 

to the Royal Society on the 27th October, 1864, contains Maxwell's mature thoughts on a 

subject which had so long occupied his mind. It was afterwards reproduced in his Treatise 

with trifling modifications in the treatment of its parts, but without substantial changes 

in its main features. In this paper Maxwell reverses the mode of treating electrical 

phenomena adopted by previous mathematical writers; for while they had sought to build 

up the laws of the subject by starting from the principles discovered by Ampere, and 

deducing the induction of currents from the conservation of energy, Maxwell adopts the 

method of first arriving at the laws of induction and then deducing the mechanical 

attractions and repulsions. 

After recalling the general phenomena of the mutual action of cuiTents and magnets 
and the induction produced in a circuit by any variation of the strength of the field m 
which it lies, the propagation of light through a luminiferous medium, the properties of 
dielectrics and other phenomena which point to a medium capable of transmittmg force 
and motio^i, he proceeds. — 

"Thus then we are led to the conception of a complicated mechanism capable 

of a vast variety of motions but at the same time so connected that the motion of 

one part depends, according to definite relations, on the motion of other parts, these 

teotions being communicated by forces arising from the relative displacement of their 

connected parts, in virtue of their elasticity. Such a mechanism must be subject 

to the laws of Dynamics." 

On applying dynamical principles to such a connected system he attains certain general 

propositions which, on being compared with the laws of induced currents, enable him to 

identify certain features of the mechanism with properties of currents. The induction of 

currehts and their electromagnetic attraction are thus explained and connected. 



XXll PREFACE. 

In a subsequent part of the memoir he proceeds to establish from these premises 
the general equations of the Field and obtains the usual formulae for the mechanical 
force on currents, magnets and bodies possessing an electrostatic charge. 

He also returns to and elaborates more fully the electromagnetic Theory of Light. 
His equations shew that dielectrics can transmit only transverse vibrations, the speed of 
propagation of which in air as deduced from electrical data comes out practically identical 
with the known velocity of light. For other dielectrics the index of refraction is equal 
to the square root of the product of the specific inductive capacity by the coefficient of 
magnetic induction, which last factor is for most bodies practically unity. Various comparisons 
have been made with the view of testing this deduction. In the case of paraffin wax and 
some of the hydrocarbons, theory and experiment agree, but this is not the case with 
glass and some other substances. Maxwell has also applied his theory to media which 
are not perfect insulators, and finds an expression for the loss of light in passing through 
a stratum of given thickness. He remarks in confirmation of his result that most good 
conductors are opaque while insulators are transparent, but he also adds that electrolytes 
which transmit a current freely are often transparent, while a piece of gold leaf whose 
resistance was determined by Mr Hockin allowed far too great an amount of light to 
pass. He observes however that it is possible "there is less loss of energy when the 
electromotive forces are reversed with the rapidity of light than when they act for sensible 
times as in our experiments." A similar explanation may be given of the discordance 
between the calculated and observed values of the specific inductive capacity. Prof. J. J, 
Thomson in the Proceedings of the Royal Society, Vol. 46, has described an experiment by 
which he has obtained the specific inductive capacities of various dielectrics when acted 
on by alternating electric forces whose frequency is 25,000,000 per second. He finds that 
under these conditions the specific inductive capacity of glass is very nearly the same as 
the square of the refractive index, and very much less than the value for slow rates of 
reversals. In illustration of these remarks may be quoted the observations of Prof. Hertz who 
has shewn that vulcanite and pitch are transparent for waves, whose periods of vibration are 
about three hundred millionths of a second. The investigations of Hertz have shewn that 
electro-dynamic radiations are transmitted in waves with a velocity, which, if not equal to, is 
comparable with that of light, and have thus given conclusive proof that a satisfactory 
theory of Electricity must take into account in some form or other the action of the 
dielectric. But this does not prove that Maxwell's theory is to be accepted in every 
particular. A peculiarity of his theory is, as he himself points out in his treatise, that 
the variation of the electric displacement is to be treated as part of the current as well 
as the current of conduction, and that it is the total amount due to the sum of these 
which flows as if electricity were an incompressible fluid, and which determines external 
electrodynamic actions. In this respect it differs from the theory of Helmholtz which 
also takes into account the action of the dielectric. Professor J. J. Thomson » in his 
Review of Electric Theories has entered into a full discussion of the points at issue 



PREFACE. XXlll 

between the two above mentioned theories, and the reader is referred to his paper for 
further information *. Maxwell in the memoir before us has also applied his theory to 
the passage of light through crystals, and gets rid at once of the wave of normal vibrations 
which has hitherto proved the stumbling block in other theories of light. 

The electromagnetic Theory of Light has received numerous developments at the hands 
of Lord Rayleigh, Mr Glazebrook, Professor J. J. Thomson and others. These volumes 
also contain various shorter papers on Electrical Science, though perhaps the most complete 
record of Maxwell's work in this department is to be found in his Treatise on Electricity 
and Magnetism in which they were afterwards embodied. 

Another series of papers of hardly less importance than those on Electricity are the 
various memoirs on the Dynamical Theory of Gases. The idea that the properties of 
matter might be explained by the motions and impacts of their ultimate atoms is as 
old as the time of the Greeks, and Maxwell has given in his paper on " Atoms " a full 
sketch of the ancient controversies to which it gave rise. The mathematical difficulties of 
the speculation however were so great that it made little real progress till it was taken 
up by Clausius and shortly afterwards by Maxwell. The first paper by Maxwell on the 
subject is entitled "Illustrations of the Dynamical Theory of Gases" and was published 
in the Philosophical Magazine for January and July, 1860, having been read at a meeting 
of the British Association of the previous year. Although the methods developed in this 
paper were afterwards abandoned for others, the paper itself is most interesting, as it indicates 
clearly the problems in the theory which Maxwell proposed to himself for solution, and so far 
contains the germs of much that was treated of in his next memoir. It is also epoch-making, 
inasmuch as it for the first time enumerates various propositions which ai-e characteristic 
of Maxwell's work in this subject. It contains the first statement of the distribution of velo- 
cities according to the law of errors. It also foreshadows the theorem that when two gases 
are in thermal equilibrium the mean kinetic energy of the molecules of each system is the 
same ; and for the first time the question of the viscosity of gases is treated dynamically. 

In his great memoir "On the Dynamical Theory of Gases" published in the Philo- 
sophical Transactions of the Royal Society and read before the Society in May, 1866, he 
returns to this subject and lays down for the first time the general d3niamical methods 
appropriate for its treatment. Though to some extent the same ground is traversed as in 
his former paper, the methods are widely different. He here abandons his former h}^othesis 
that the molecules are hard elastic spheres, and supposes them to repel each other with 
forces varying inversely as the fifth power of the distance. His chief reason for assuming 
this law of action appears to be that it simplifies considerably the calculation of the 
collisions between the molecules, and it leads to the conclusion that the coefficient of 
viscosity is directly proportional to the absolute temperature. He himself undertook an 
experimental enquiry for the purpose of verifying this conclusion, and, in his paper on the 
Viscosity of Gases, he satisfied himself of its correctness. A re-examination of the numerical 

* British Association Report, 1885. 



XXIV PREFACE. 

reductions made in the course of his work discloses however an inaccuracy which materially 
affects the values of the coefl&cient of viscosity obtained. Subsequent experiments also seem 
to shew that the concise relation he endeavoured to establish is by no means so near 
the truth as he supposed, and it is more than doubtful whether the action between two 
molecules can be represented by any law of so simple a character. 

In the same memoir he gives a fresh demonstration of the law of distribution of 
velocities, but though the method is of permanent value, it labours under the defect of 
assuming that the distribution of velocities in the neighbourhood of a point is the same 
in every direction, whatever actions may be taking place within the gas. This flaw in 
the argument, first pointed out by Boltzmann, seems to have been recognised by Maxwell, 
who in his next paper "On the Stresses in Rarefied Gases arising from inequalities of 
Temperature," published in the Philosophical Transactions for 1879, Part I., adopts a form 
of the distribution function of a somewhat different shape. The object of this paper was 
to arrive at a theory of the effects observed in Crookes's Radiometer. The results of the 
investigation are stated by Maxwell in the introduction to the paper, from which it would 
appear that the observed motion cannot be explained on the Dynamical Theory, unless it 
be supposed that the gas in contact with a solid can slide along the surface with a finite 
velocity between places whose temperatures are different. In an appendix to the paper 
he shews that on certain assumptions regarding the nature of the contact of the solid 
and gas, there will be, when the pressure is constant, a flow of gas along the surface 
from the colder to the hotter parts. The last of his longer papers on this subject is 
one on Boltzmann's Theorem. Throughout these volumes will be found numerous shorter 
essays on kindred subjects, published chiefly in Nature and in the Encyclopcedia Britannica. 
Some of these contain more or less popular expositions of this subject which Maxwell 
bad himself in great part created, while others deal with the work of other writers in 
the same field. They are profoundly suggestive in almost every page, and abound in acute 
criticisms of speculations which he could not accept. They are always interesting; for 
although the larger papers are sometimes difficult to follow, Maxwell's more popular writings 
are characterized by extreme lucidity and simplicity of style. 

The first of Maxwell's papers on Colour Perception is taken from the Transactions of 
the Royal Scottish Society of Arts and is in the form of a letter to Dr G. Wilson dated 
Jan. 4, 1855. It was followed directly afterwards by a communication to the Royal Society 
of Edinburgh, and the subject occupied his attention for some years. The most important 
of his subsequent work is to be found in the papers entitled "An account of Experiments 
on the Perception of Colour " published in the Philosophical Magazine, Vol xiv. and " On 
the Theory of Compound Colours and its relation to the colours of the spectrum " in the 
Philosophical Transactions for the year 1860. We may also refer to two lectures delivered 
at the Royal Institution, in which he recapitulates and enforces his main positions in his 
usual luminous style. Maxwell from the first adopts Young's Theory of Colour Sensation, 
according to which all colours may ultimately be reduced to three, a red, a green and 



PREFACE. XXV 

a violet. This theory had been revived by Helmholtz who endeavoured to find for it a 
physiological basis. Maxwell however devoted himself chiefly to the invention of accurate 
methods for combining and recording mixtures of colours. His first method of obtaining 
mixtures, that of the Colour Top, is an adaptation of one formerly employed, but in 
Maxwell's hands it became an instrument capable of giving precise numerical results by 
means which he added of varying and measuring the amounts of colour which were 
blended in the eye. In the representation of colours diagrammatical ly he followed Young 
in employing an equilateral triangle at the angles of which the fundamental colours were 
placed. All colours, white included, which may be obtained by mixing the fundamental 
colours in any proportions will then be represented by points lying within the triangle. 
Points without the triangle represent colours which must be mixed with one of the funda- 
mental tints to produce a mixture of the other two, or with which two of them must be 
mixed to produce the third. 

In his later papers, notably in that printed in the Philosophical Transactions, he 
adopts the method of the Colour Box, by which different parts of the spectrum may be 
mixed in different proportions and matched with white, the intensity of which has been 
suitably diminished. In this way a series of colour equations are obtained which can be 
used to evaluate any colour in terms of the three fundamental colours. These observations 
on which Maxwell expended great care and labour, constitute by far the most important 
data regarding the combinations of colour sensations which have been yet obtained, and 
are of permanent value whatever theory may ultimately be adopted of the physiology of the 
perception of colour. 

In connection with these researches into the sensations of the normal eye, may be 
mentioned the subject of colour-blindness, which also engaged Maxwell's attention, and is 
discussed at considerable length in several of his papers. 

Geometrical Optics was another subject in which Maxwell took much interest. At an early 
period of his career he commenced a treatise on Optics, which however was never completed. 
His first paper "On the general laws of optical instruments," appeared in 1858, but a brief 
account of the first part of it had been previously communicated to the Cambridge Philosophical 
Society. He therein lays down the conditions which a perfect optical instrument must fulfil, 
and shews that if an instrument produce perfect images of an object, i.e. images free from 
astigmatism, curvature and distortion, for two different positions of the object, it will give 
perfect images at all distances. On this result as a basis, he finds the relations between 
the foci of the incident and emergent pencils, the magnifying power and other characteristic 
quantities. The subject of refraction through optical combinations was afterwards treated 
by him in a different manner, in three papers communicated to the London Mathematical 
Society. In the first (1873), "On the focal lines of a refracted pencil," he applies Hamilton's 
characteristic function to determine the focal lines of a thin pencil refracted from one 
isotropic medium into another at any surface of separation. In the second (1874), "On 



XXVI PREFACE. 

Hamilton's characteristic function for a narrow beam of light," he considers the more general 
question of the passage of a ray from one isotropic medium into another, the two media 
being separated by a third which may be of a heterogeneous character. He finds the most 
general form of Hamilton's characteristic function from one point to another, the first being 
in the medium in which the pencil is incident and the second in the medium in which 
it is emergent, and both points near the principal ray of the pencil. This result is then 
applied in two particular cases, viz. to determine the emergent pencil (1) from a spectroscope, 
(2) from an optical instrument symmetrical about its axis. In the third paper (1875) he 
resumes the last-mentioned application, discussing this case more fully under a somewhat 
simplified analysis. 

It may be remarked that all these papers are connected by the same idea, which was — 
first to study the optical efiects of the entire instrument without examining the mechanism 
by which these effects are produced, and then, as in the paper in 1858, to supply whatever 
data may be necessary by experiments upon the instrument itself. 

Connected to some extent with the above papers is an investigation which was published 
in 1868 " On the cyclide." As the name imports, this paper deals chiefly with the geometrical 
properties of the surface named, but other matters are touched on, such as its conjugate 
isothermal functions. Primarily however the investigation is on the orthogonal surfaces to 
a system of rays passing accurately through two lines. In a footnote to this paper Maxwell 
describes the stereoscope which he invented and which is now in the Cavendish Laboratory. 

In 1868 was also published a short but important article entitled " On the best arrange- 
ment for producing a pure spectrum on a screen." 

The various papers relating to the stresses experienced by a system of pieces joined 
together so as to form a frame and acted on by forces form an important group connected 
with one another. The first in order was "On reciprocal figures and diagrams of forces," 
published in 1864. It was immediately followed by a paper on a kindred subject, "On 
the calculation of the equilibrium and stiffness of frames." In the first of these Maxwell 
demonstrates certain reciprocal properties in the geometry of two polygons which are related 
to one another in a particular way, and establishes his well-known theorem in Graphical 
Statics on the stresses in frames. In the second he employs the principle of work to 
problems connected with the stresses in frames and structures and with the deflections 
arising from extensions in any of the connecting pieces. 

A third paper " On the equilibrium of a spherical envelope," published in 1867, may 
here be referred to. The author therein considers the stresses set up in the envelope by 
a system of forces applied at its surface, and ultimately solves the problem for two normal 
forces applied at any two points. The solution, in which he makes use of the principle 
of inversion as it is applied in various electrical questions, turns ultimately on the deter- 
mination of a certain function first introduced by Sir George Airy, and called by Maxwell 



PREFACE. XXvii 

Airy's Function of Stress. The methods which in this paper were attended with so much 
success, seem to have suggested to Maxwell a reconsideration of his former work, with the 
view of extending the character of the reciprocity therein established. Accordingly in 1870 
there appeared his fourth contribution to the subject, "On reciprocal figures, frames and 
diagrams of forces." This important memoir was published in the Transactions of the Royal 
Society of Edinburgh, and its author received for it the Keith Prize. He begins with a 
remarkably beautiful construction for drawing plane reciprocal diagrams, and then proceeds 
to discuss the geometry and the degrees of freedom and constraint of polyhedral frames, 
his object being to lead up to the limiting case when the faces of the polyhedron become 
infinitely small and form parts of a continuous surface. In the course of this work he 
obtains certain results of a general character relating to inextensible surfaces and certain 
otjiers of practical utility relating to loaded frames. He then attacks the general problem of 
representing graphically the internal stress of a body and by an extension of the meaning 
of "Diagram of Stress," he gives a construction for finding a diagram which has mechanical 
as well as geometrical reciprocal properties with the figure supposed to be under stress. It 
is impossible with brevity to give an account of this reciprocity, the development of which 
in Maxwell's hands forms a very beautiful example of analysis. It will be suflScient to 
state that under restricted conditions this diagram of stress leads to a solution for the 
components of stress in terms of a single function analogous to Airy's Function of Stress. 
In the remaining parts of the memoir there is a discussion of the equations of stress, and 
it is shewn that the general solution may be expressed in terms of three functions analogous 
to Airy's single function in two dimensions. These results are then applied to special 
cases, and in particular the stresses in a horizontal beam with a uniform load on its upper 
surface are fully investigated. 

On the subjects in which Maxwell's investigations were the most numerous it has 
been thought necessary, in the observations which have been made, to sketch out briefly 
the connections of the various papers on each subject with one another. It is not how- 
ever intended to enter into an account of the contents of his other contributions to science, 
and this is the less necessary as the reader may readily obtain the information he may 
require in Maxwell's own language. It was usually his habit to explain by way of 
introduction to any paper his exact position with regard to the subject matter and to 
give a brief account of the nature of the work he was contributing. There are however 
several memoirs which though unconnected with others are exceedingly interesting in them- 
selves. Of these the essay on Saturn's Rings will probably be thought the most important 
as containing the solution of a diflScult cosmical problem ; there are also various papers on 
Dynamics, Hydromechanics and subjects of pure mathematics, which are most useful con- 
tributions on the subjects of which they treat. 

The remaining miscellaneous papers may be classified under the following heads: (a) 
Lectures and Addresses, (b) Essays or Short Treatises, (c) Biographical Sketches, (d) Criticisms 
and Reviews. 



XXVIU PREFACE. 

Class (a) comprises his addresses to the British Association, to the London Mathematical 
Society, the Rede Lecture at Cambridge, his address at the opening of the Cavendish 
Laboratory and his Lectures at the Royal Institution and to the Chemical Society. 

Class (6) includes all but one of the articles which he contributed to the Encyclo- 
pcedia Britanrdca and several others of a kindred character to Nature. 

Class (c) contains such articles as " Fai-aday " in the Encyclopcedia Britannica and 
" Helmholtz " in Nature. 

Class (d) is chiefly occupied with the reviews of scientific books as they were pub- 
lished. These appeared in Nature and the most important have been reprinted in these 
pages. 

In some of these writings, particularly those in class (b), the author allowed himself a 
gi-eater latitude in the use of mathematical symbols and processes than in others, as 
for instance in the article " Capillary Attraction," which is in fact a treatise on that subject 
treated mathematically. The lectures were upon one or other of the three departments 
of Physics with which he had mainly occupied himself; — Colour Perception, Action through 
a Medium, Molecular Physics; and on this account they are the more valuable. In the 
whole series of these more popular sketches we find the same clear, graceful delineation of 
principles, the same beauty in arrangement of subject, the same force and precision in 
expounding proofs and illustrations. The style is simple and singularly free fi-om any kind 
of haze or obscurity, rising occasionally, as in his lectures, to a strain of subdued eloquence 
when the emotional aspects of the subject overcome the purely speculative. 

The books which were written or edited by Maxwell and published in his lifetime but 
which are not included in this collection were the "Theory of Heat" (1st edition, 1871); 
"Electricity and Magnetism" (1st edition, 1873); "The Electrical Researches of the Hon- 
ourable Henry Cavendish, F.R.S., written between 1771 and 1781, edited from the original 
manuscripts in the possession of the Duke of Devonshire, K.G." (1879). To these may be 
added a graceful little introductory treatise on Dynamics entitled "Matter and Motion" 
(published in 1876 by the Society for promoting Christian Knowledge). Maxwell also 
contributed part of the British Association Report on Electrical Units which was afterwards 
published in book form by Fleeming Jenkin. 

The "Theory of Heat" appeai-ed in the Text Books of Science series published by 
Longmans, Green and Co., and was at once hailed as a beautiful exposition of a subject, 
part of which, and that the most interesting part, the mechanical theory, had as yet but 
commenced the existence which it owed to the genius and laboui-s of Rankine, Thomson and 
Clausius. There is a certain charm in Maxwell's treatise, due to the freshness and originality 
of its expositions which has rendered it a great favourite with students of Heat. 

After his death an " Elementary Treatise on Electricity," the greater part of which he 
had written, was completed by Professor Garnett and published in 1881. The aim of this 



PREFACE. XXIX 

treatise and its position relatively to his larger work may be gathered from the following 
extract from Maxwell's preface. 

" In this smaller book I have endeavoured to present, in as compact a form as I 
can, those phenomena which appear to throw light on the theory of electricity and to 
use them, each in its place, for the development of electrical ideas in the mind of 
the reader." 

"In the larger treatise I sometimes made use of methods which I do not think 
the best in themselves, but without which the student cannot follow the investigations 
of the founders of the Mathematical Theory of Electricity. I have since become more 
convinced of the superiority of methods akic to those of Faraday, and have therefore 
adopted them from the first." 

Of the "Electricity and Magnetism" it is difficult to predict the future, but there is 
no doubt that since its publication it has given direction and colour to the study of 
Electrical Science. It was the master's last word upon a subject to which he had devoted 
several years of his life, and most of what he wrote found its proper place in the treatise. 
Several of the chapters, notably those on Electromagnetism, are practically reproductions of 
his memoirs in a modified or improved form. The treatise is also remarkable for the handling 
of the mathematical details no less than for the exposition of physical principles, and is 
enriched incidentally by chapters of much originality on mathematical subjects touched on 
in the course of the work. Among these may be mentioned the dissertations on Spherical 
Harmonics and Lagrange's Equations in Dj-namics. 

The origin and growth of Maxwell's ideas and conceptions of electrical action, cul- 
minating in his treatise where all these ideas are arranged in due connection, form an 
interesting chapter not only in the history of an individual mind but in the history of 
electrical science. The importance of Faraday's discoveries and speculations can hardly be 
overrated in their influence on Maxwell, who tells us that before he began the study of 
electricity he resolved to read none of the mathematics of the subject till he had first 
mastered the "Experimental Researches." He was also at first under deep obligations to 
the ideas contained in the exceedingly important papers of Sir W. Thomson on the analogy 
between Heat-Conduction and Statical Electricity and on the Mathematical Theory of 
Electricity in Equilibrium. In his subsequent efforts we must perceive in Maxwell, possessed 
of Faraday's views and embued with his spirit, a vigorous intellect bringing to bear on a 
subject still full of obscurity the steady light of patient thought and expending upon it 
all the resources of a never failing ingenuity. 

Royal Navax College, 
Greenwich, 

August, 1890. 



TABLE OF CONTENTS. 



II. 
Ill 

IV. 

V. 

VI. 

VII. 

IX. 

X. 
XI. 

XII. 
XIII. 

XIV. 
XV. 




On the Description of Oval Curves and those having a plurality of Foci; with 

remarks by Professor Forbes 

On the Theory of Rolling Curves ■* 

On the Equilibrium of Elastic Solids ^^ 

Solutions of Problems 

On the Transformation of Surfaces by Bending 80 

On a paHicular case of the descent of a heavy body in a resisting medium . 115 

On the Theory of Colours in relation to Colour- Blindness 119 

Experiments on Colour as perceived by the Eye, with remarks on Colour -Blindness 126 

On Faraday's Lines of Force ^"^^ 

Description of a New Form of the Platometer, an Instrument for measuring the 

areas of Plane Figures drawn on paper 230 

On the elementary theory of Optical Instruments 238 

On a method of drawing the Theoretical Form3 of Faraday's Lines of Force 

without calculation 

On the unequal sensibility of the Foramen Centrale to Light of different Colours 242 
On the Theory of Compound Colours with reference to mixtures of Blue and 

Yellow Light ^^'^ 

On an instrument to illustrate Poimot's TJieory of Rotation . . • .246 
On a Dynamical Top, for exhibiting the phenomena of the motions of a body of 
invariable form about a fixed point, with s&ine suggestions as to the Earth's 

motion 

Account of Experiments on the Perception of Colour 263 

97 1 
On the general laius of Optical Instruments 

On Theories of the Constitution of Saturn's Rings 286 

On the stability of the motion of Saturn s Rings 288 

Illustrations of the Dynamical Theory of Gases 

On the Theory of Compound Colours and the Relations of the Colours of the 
Spectrum 

On the Theory of Three Primary Colours ***^ 

451 

On Physical Lines of Force 

On Reciprocal Figures and Diagrams of Forces °^* 

A Dynamical Theory of the Electromagnetic Field 526 

On the Calculation of the EquilibHum and Stiffness of Frames .... 598 



ERRATA. 



Page 40. In the first of equations (12), second group of terms, read 
(hP dy' d^ 



instead of 



d^^^d^^^d^^ 



with corresponding changes in the other two equations. 

Page 153, five lines from bottom of page, read 127 instead of 276 

Page 591, four lines from bottom of page the equation should be 

d^M d2M_ldM 
da? "^ db' a da~ 

Page 592, in the first line of the expression for L change 

- K cos 26 into - ^ cosec 26. 



[From the Proceedings of the Royal Society of Edinburgh, Vol, li. April, 1846.] 



I. On the Description of Oval Curves, and those having a plurality of Foci; ivith 
remarks by Professor Forbes. Communicated by Professor Forbes. 



Mr Clerk Maxwell ingeniously suggests the extension of the common 
theory of the foci of the conic sections to curves of a higher degree of com- 
plication in the following manner : — 

(1) As in the ellipse and hyperbola, any point in the curve has the 
sum or difference of two lines drawn from two points or foci = a. constant 
quantity, so the author infers, that curves to a certain degree analogous, may 
be described and determined by the condition that the simple distance from 
one focus pliLS a multiple distance from the other, may be = a constant quantity; 
or more generally, m times the one distance + n times the other = constant. 

(2) The author devised a simple mechanical means, by the wrapping 
of a thread round pins, for producing these curves. See Figs. 1 and 2. He 



Fig. 1. Two FocL Katios 1, 



Fig. 2. Two Foci Ratios 2, 3. 




then thought of extending the principle to other curves, whose property 
should be, that the sum of the simple or multiple distances of any point of 



DESCRIPTION OF OVAL CURVES. 



the curve from three or more points or foci, should be = a constant quantity ; 
and this, too, he has effected mechanically, by a very simple arrangement of 
a string of given length passing round three or more fixed pins, and con- 
straining a tracing point, P. See Fig. 3. Farther, the author regards curves 

Fig. 3. Three Foci. Eatios of Equality. 




of the first kind as constituting a particular class of curves of the second 
kind, two or more foci coinciding in one, a focus in which two strings meet 
being considered a double focus; when three strings meet a treble focus, &c. 

Professor Forbes observed that the equation to curves of the first class is 
easily found, having the form 

V^+7= a-VhJ{x- c)' + y\ 

which is that of the curve known under the name of the First Oval of 
Descartes*. Mr Maxwell had already observed that when one of the foci was 
at an infinite distance (or the thread moved parallel to itself, and was confined 
in respect of length by the edge of a board), a curve resembling an ellipse 
was traced ; from which property Professor Forbes was led first to infer the 
identity of the oval with the Cartesian oval, which is well known to have this 
property. But the simplest analogy of all is that derived from the method of 
description, r and r being the radients to any point of the curve from the two 

foci ; 

mr + nr — constant, 

which in fact at once expresses on the undulatory theory of light the optical 
character of the surface in question, namely, that light diverging from one 
focus F without the medium, shall be correctly convergent at another point / 

* Herschel, On Light, Art. 232 ; Lloyd, On Light and Vision, Chap. vii. 



DESCRIPTION OF OVAL CURVES. J 

within it ; and in this case the ratio — expresses the index of refraction of 

the medium*. 

If we denote by the power of either focus the number of strings leading 
to it by Mr Maxwell's construction, and if one of the foci be removed to an 
infinite distance, if the powers of the two foci be equal the curve is a parabola ; 
if the power of the nearer focus be greater than the other, the curve is an 
eUipse; if the power of the infinitely distant focus be the greater, the curve 
is a hyperbola. The first case evidently corresponds to the case of the reflection 
of parallel rays to a focus, the velocity being unchanged after reflection; the 
second, to the refraction of parallel rays to a focus in a dense medium (in 
which light moves slower) ; the third case to refraction into a rarer medium. 

The ovals of Descartes were described in his Geometry, where he has also 
given a mechanical method of describing one of themt, but only in a particular 
case, and the method is less simple than Mr Maxwell's. The demonstration of 
the optical properties was given by Newton in the Principia, Book i., prop. 97, 
by the law of the sines; and by Huyghens in 1690, on the Theory of Undu- 
lations in his Traite de la Lumiere. It probably has not been suspected that 
so easy and elegant a method exists of describing these curves by the use of 
a thread and pins whenever the powers of the foci are commensurable. For 
instance, the curve. Fig. 2, drawn with powers 3 and 2 respectively, give the 
proper form for a refracting surface of a glass, whose index of refraction is 1'50, 
in order that rays diverging from f may be refracted to F. 

As to the higher classes of curves with three or more focal points, we 
cannot at present invest them with equally clear and curious physical properties, 
but the method of drawing a curve by so simple a contrivance, which shall 
satisfy the condition 

mr + nr +pr" + &c. = constant, 

is in itself not a little interesting; and if we regard, with Mr Maxwell, the 
ovals above described, as the limiting case of the others by the coalescence 
of two or more foci, we have a farther generalization of the same kind as that 
so highly recommended by Montucla^ by which Descartes elucidated the conic 
sections as particular cases of his oval curves. 

♦ This was perfectly well shewn by Hnyghens in his Traite de la Lumiere, p. 111. (1690.) 

+ Edit. 1683. Geometria, Lib. ii. p. 54. 

X Histoire dea Mathematiqties. First Edit IL 102. 



[From the Transactions of the Royal Society of Edinburgh, Vol. xvi. Part v.] 



II. On the Theory of Rolling Curves. Communicated by the Eev. Professor 

Kelland. 



There is an important geometrical problem which proposes to find a curve 
having a given relation to a series of curves described according to a given 
law. This is the problem of Trajectories in its general form. 

The series of curves is obtained from the general equation to a curve by 
the variation of its parameters. In the general case, this variation may change 
the form of the curve, but, in the case which we are about to consider, the 
curve is changed only in position. 

This change of position takes place partly by rotation, and partly by trans- 
ference through space. The roUing of one curve on another is an example of 
this compound motion. 

As examples of the way in which the new curve may be related to the 
series of curves, we may take the following : — 

1. The new curve may cut the series of curves at a given angle. When 
this angle becomes zero, the curve is the envelope of the series of curves. 

2. It may pass through correspondiug points in the series of curves. 
There are many other relations which may be imagined, but we shall confine 
our attention to this, partly because it aSbrds the means of tracing various 
curves, and partly on account of the connection which it has with many 
geometrical problems. 

Therefore the subject of this paper will be the consideration of the relations 
of three curves, one of which is fixed, while the second rolls upon it and 
traces the third. The subject of rolling curves is by no means a new one. 
The first idea of the cycloid is attributed to Aristotle, and involutes and 
evolutes have been long known. 



THE THEORY OF ROLLING CURVES. 

In the Histmy of the Royal Academy of Sciences for 1704, page 97, 
there is a memoir entitled "Nouvelle formation des Spirales," by M. Varignon, 
in which he shews how to construct a polar curve from a curve referred to 
rectangular co-ordinates by substituting the radius vector for the abscissa, and 
a circular arc for the ordinate. After each curve, he gives the curve into 
which it is " unrolled," by which he means the curve which the spiral must 
be rolled upon in order that its pole may trace a straight line; but as this 
18 not the principal subject of his paper, he does not discuss it very fully. 

There is also a memoir by M. de la Hire, in the volume for 1706, Part ii., 
page 489, entitled "Methode generale pour r^duire toutes les Lignes courbes ^ 
des Roulettes, leur generatrice ou leur base ^tant donnde telle qu'on voudra." 

M. de la Hire treats curves as if they were polygons, and gives geome- 
trical constructions for finding the fixed curve or the rolling curve, the other 
two being given; but he does not work any examples. 

In the volume for 1707, page 79, there is a paper entitled, "Methode 
generale pour determiner la nature des Courbes form^es par le roulement de 
toutes sortes de Courbes sur une autre Courbe quelconque." Par M. Nicole. 

M. Nicole takes the equations of the three curves referred to rectangular 
co-ordinates, and finds three general equations to connect them. He takes the 
tracing-point either at the origin of the co-ordinates of the rolled curve or not. 
He then shews how these equations may be simplified in several particular 
cases. These cases are — 

(1) When the tracing-point is the origin of the roUed curve. 

(2) When the fixed curve is the same as the rolling cxirve. 

(3) When both of these conditions are satisfied. 

(4) When the fixed line is straight. 

He then says, that if we roll a geometric curve on itself, we obtain a new 
geometric curve, and that we may thus obtain an infinite number of geometric 
curves. 

The examples which he gives of the application of his method are all taken 
from the cycloid and epicycloid, except one which relates to a parabola, rolling 
on itself, and tracing a cissoid with its vertex. The reason of so small a 
number of examples being worked may be, that it is not easy to eliminate 
the co-ordinates of the fixed and rolling curves from his equations. 

The case in which one curve roUing on another produces a circle is treated 
of in Willis's Principles of Mechanism. Class C. Boiling Contact. 



6 THE THEORY OP ROLLHiTO CURVES. 

He employs the same method of finding the one curve from the other 
which is used here, and he attributes it to Euler (see the Acta Petropolitana, 
Vol. v.). 

Thus, nearly all the simple cases have been treated of by different authors; 
but the subject is still far from being exhausted, for the equations have been 
applied to very few curves, and we may easily obtain new and elegant proper- 
ties from any curve we please. 

Almost all the more notable curves may be thus linked together in a great 
variety of ways, so that there are scarcely two curves, however dissimilar, 
between which we cannot form a chain of connected curves. 

This will appear in the list of examples given at the end of this paper. 




Let there be a curve KAS, whose pole is at C. 



THE THEORY OF ROLLING CURVES. 7 

Let the angle DCA = 6, and CA=r, and let 

Let this curve remain fixed to the paper. 

Let there be another curve BAT, whose pole is B. 

Let the angle MBA = 0t, and BA=r^, and let 

Let this curve roll along the curve KAS without slipping. 
Then the pole B will describe a third curve, whose pole is C. 
Let the angle DCB = 0^, and CB = r„ and let 

We have here six unknown quantities 0,dAr,r^r^; but we have only three 
equations given to connect them, therefore the other three must be sought for 
in the enunciation. 

But before proceeding to the investigation of these three equations, we must 
premise that the three curves will be denominated as follows : — 
The Fixed Curve, Equation, e^ = ^^{r^. 
The Rolled Curve, Equation, 0. = <f>,{r,). 
Tlie Traced Curve, Equation, 6^ = 4>.,{r^. 

When it is more convenient to make use of equations between rectangular 
co-ordinates, we shall use the letters x^^, x^^, x^ij^. We shall always employ the 
letters s^s^^ to denote the length of the curve from the pole, p.p^p^ for the per- 
pendiculars from the pole on the tangent, and q^q/i^ for the intercepted part of 
the tangent. 

Between these quantities, we have the following equations: — 

r = ^/^T?, ^ = tan-|, 

a? = r cos ^, y = r sin 6, 

r" ydx — xdy 



jm'S ""^w+w' 



THE THEORY OF ROLLING CURVES. 

rdr 

dS _ xdx + ydy 



2=-r=7x!fi' r- 



J{dxy + (dyY' 



' "^ W '^d^ daf 

We come now to consider the three equations of rolling which are involved 
in the enunciation. Since the second curve rolls upon the first without slipping, 
the length of the fixed curve at the point of contact is the measure of the 
length of the rolled curve, therefore we have the following equation to connect 
the fixed curve and the rolled curve — 

«! = Sj. 

Now, by combining this equation with the two equations 

it is evident that from any of the four quantities 6{r^6^r^ or x^^x^^, we can 
obtain the other three, therefore we may consider these quantities as known 
functions of each other. 

Since the curve rolls on the fixed curve, they must have a common tangent. 

Let PA be this tangent, draw BP, CQ perpendicular to PA, produce CQ, 
and draw BR perpendicular to it, then we have CA=r^, BA = r^, and CB = r,; 
CQ=p„ PB=p,, and BN=p,; AQ = q„ AP = q„ and CN=q,. 

Also r,'=CR=CR + RR = (CQ + PBY+(AP-AQf 

=p,' + 2p,p, +p,' + r,' -p,' - 2q,q, + r," -p,' 
fz = n' + n' + 2piPa - 2q,q^. 
Since the first curve is fixed to the paper, we may find the angle 6,. 
Thus e, = DCB = DCA + ACQ + RCB 

= e?. + tan-| + tan-|§ 
^, = ^, + tan--^ + tan-^ ^^^^ 



TjdO^ Pi +pi 



THE THEORY OF ROLLING CURVES. » 

Thus we have found three independent equations, which, together with the 
equations of the curves, make up six equations, of which each may be deduced 
from the others. There is an equation connecting the radii of curvature of the 
three curves which is sometimes of use. 

The angle through which the rolled curve revolves during the description of 
the element ds„ is equal to the angle of contact of the fixed curve and the 
rolling curve, or to the sum of their curvatures, 

ds^ ds^ ds. 

But the radius of the rolled curve has revolved In the opposite direction 
through an angle equal to dO,, therefore the angle between two successive posi- 
tions of r, is equal to -^-dd,. Now this angle is the angle between two 

successive positions of the normal to the traced curve, therefore, if be the 
centre of curvature of the traced curve, it is the angle which ds^ or ds^ subtends 
at 0. Let OA^T, then 

ds^ r4d^ ds, ,^ _ ds^ ds, ,. 

^J__J_ 1 _^ 
•*• '^'ds, T~ R, R, ds/ 

-tAt^tJ RJR.' 

As an example of the use of this equation, we may examine a property 
of the logarithmic spiral. 

In this curve, p = mr, and R = — , therefore if the rolled curve be the 
■^ m 

logarithmic spiral 

/I 1\ 1 ^m 

"^[t^tJ-r^v/ 

m_ 1 

t~r:,* 

AO 
therefore ^0 in the figure = ?ni2i, and -^ = m. 

Let the locus of 0, or the evolute of the traced curve LYBH, be the 
curve OZY, and let the evolute of the fixed curve KZAS be FEZ, and let 
us consider FEZ as the fixed curve, and OZF as the traced curve. 



10 THE THEORY OF ROLLING CURVES. 

Then in the triangles BPA, AOF, we have OAF=PBA, and ^='^ = ^y 

therefore the triangles are similar, and FOA = APB = - , therefore OF is perpen- 
dicular to OA, the tangent to the curve OZY, therefore OF is the radius of 
the curve which when roUed on FEZ traces OZY, and the angle which the 
curve makes with this radius is OFA=PAB = %mr^m, which is constant, there- 
fore the curve, which, when rolled on FEZ, traces OZY, is the logarithmic 
spiral. Thus we have proved the following proposition : " The involute of the 
curve traced by the pole of a logarithmic spiral which rolls upon any curve, 
is the curve traced by the pole of the same logarithmic spiral when rolled on 
the involute of the primary curve." 

It follows from this, that if we roll on any curve a curve having the 
property _2:»i — Wjri, and roll another curve having Pi = 'm^r^ on the curve traced, 
and so on, it is immaterial in what order we roll these curves. Thus, if we 
roll a logarithmic spiral, in which jp = mr, on the nth involute of a circle whose 
radius is a, the curve traced is the w+lth involute of a circle whose radius 
is Jl-m\ 

Or, if we roll successively m logarithmic spirals, the resulting curve is the 
n + mth involute of a circle, whose radius is 

aJl—m^ sll- m/, Jkc. 

We now proceed to the cases in which the solution of the problem may 
be simplified. This simplification is generally effected by the consideration that 
the radius vector of the rolled curve is the normal drawn from the traced 
curve to the fixed curve. 

In the case in which the curve is rolled on a straight line, the perpen- 
dicular on the tangent of the rolled curve is the distance of the tracing point 
from the straight line ; therefore, if the traced curve be defined by an equation 
in iCg and y„ 

'^.°p.= / "'„... (1)' 



and 



'••=^'^©^ ^'^- 



THE THEORY OF ROLLING CURVES. 11 

By substituting for r, in the first equation, its value, as derived from the 
second, we obtain 

-■©■[©■-]=©'■ 

If we know the equation to the rolled curve, we may find (-7-^') in 

terms of r,, then by substituting for r, its value in the second equation, we 

dx (1 X 

have an equation containing x^ and -^, from which we find the value of -t— ' 

dy, du, 

in terms of x^; the integration of this gives the equation of the traced curve. 

As an example, we may find the curve traced by the pole of a hyperbolic 
spiral which rolls on a straight line. 

a 

fdrA' _ rl 
,ddj ~ a' 



The equation of the rolled curve is 6^ = 



- •■©■-■[(IJ-]' 



dx^ _ ^3 
'* dy,~Ja'-x,'' 

This is the differential equation of the tractory of the straight line, which 
is the curve traced by the pole of the hyperbolic spiral. 
By eliminating x^ in the two equations, we obtain 

dr^_ /dxA 

This equation serves to determine the rolled curve when the traced cuive 
is given. 

As an example we shall find the curve, which being rolled on a straight 
line, traces a common catenary. 

Let the equation to the catenary be 



'l(e' + e-^. 



12 THE THEORY OF ROLLING CURVES. 



Then 



dy,~N a' ' 



dr 



then by integration ^ =cos'^ ( 1 j 



2a 
r = 



1+COS0' 



This is the polar equation of the parabola, the focus being the pole ; there- 
fore, if we roll a parabola on a straight line, its focus will trace a catenary. 

The rectangiilar equation of this parabola is af = Aay, and we shall now 
consider what curve must be rolled along the axis of y to trace the parabola. 

By the second equation (2), 

n = ^9 /-4- + l> but x^^Pi, 
V ^» 

.-. r/=^/ + 4a", 
.-. 2a = Vr/-jp/ = g'„ 
but q^ is the perpendicular on the normal, therefore the normal to the curve 
always touches a circle whose radius is 2a, therefore the curve is the involute 
of this circle. 

Therefore we have the following method of describing a catenary by con- 
tinued motion. 

Describe a circle whose radius is twice the parameter of the catenary; roll a 
straight line on this circle, then any point in the line will describe an involute 



THE THEORY OF ROLLING CURVES. 13 

of the circle ; roll this curve on a straight line, and the centre of the circle will 
describe a parabola ; roll this parabola on a straight line, and its focus will trace 
the catenary required. 

We come now to the case in which a straight line rolls on a curve. 

When the tracing-point is in the straight line, the problem becomes that 
of involutes and evolutes, which we need not enter upon ; and when the tracmg- 
point is not in the straight line, the calculation is somewhat complex; we shall 
therefore consider only the relations between the curves described in the first 
and second cases. 

Definition. — The curve which cuts at a given angle all the circles of a 
given radius whose centres are in a given curve, is called a tractory of the 
given curve. 

Let a straight line roll on a curve A, and let a point in the straight 
line describe a curve B, and let another point, whose distance from the first 
point is b, and from the straight line a, describe a curve C, then it is evident 
that the curve B cuts the circle whose centre is in C, and whose radius is b, 

at an angle whose sine is equal to r, therefore the curve 5 is a tractory of 

the curve C. 

When a = b, the curve B is the orthogonal tractory of the curve C. If 
tangents equal to a be drawn to the curve B, they will be terminated in 
the curve C; and if one end of a thread be carried along the curve C, the 
other end will trace the curve B. 

When a = 0, the curves B and C are both involutes of the curve A, 
they are always equidistant from each other, and if a circle, whose radius is 
6, be rolled on the one, its centre will trace the other. 

If the curve A is such that, if the distance between two points measured 
along the curve is equal to 6, the two points are similarly situate, then the 
curve B is the same with the curve C. Thus, the curve A may be a re- 
entrant curve, the circumference of which is equal to 6. 

When the curve -4 is a circle, the curves B and C are always the same. 

The equations between the radii of curvature become 

1 1 _ r 



14 THE THEORY OF ROLLING CURVES. 

When a = 0, T=0, or the centre of curvature of the curve B is at the 
point of contact. Now, the normal to the curve C passes through this point, 
therefore — 

"The normal to any curve passes through the centre of curvature of its 
tractory," 

In the next case, one curve, by rolling on another, produces a straight 
line. Let this straight line be the axis of y, then, since the radius of the 
rolled curve is perpendicular to it, and terminates in the fixed curve, and 
since these curves have a common tangent, we have this equation, 

If the equation of the rolled curve be given, find -j-^ in terms of r^, sub- 
stitute Xi for r^, and multiply by x^, equate the result to -^ , and integrate. 

Thus, if the equation of the rolled curve be 

d = Ar-"" + &c. + Kr-^ + Lr'^ + if log r + iVr + &c. + Zr"", 

^ = - n^r-(»+^) - &c. - 2Kr-' - I/p-' + Mr'' + N+ &c. + wZr"-^ 
dr 

-r-= - nAx~'* - &c. - 2Kx~"- - Lx~^ + M+ Nx + &c. + nZx", 
ax 

y = -^ Aa^-"" + &c. + 2Kx-' -L\ogx + Mx + ^Naf + &c. + -^ Zx""^', 

which is the equation of the fixed curve. 

If the equation of the fixed curve be given, find -^ in terms of cc, sub- 
stitute r for X, and divide by r, equate the result to -t-, and integrate. 

Thus, if the fixed curve be the orthogonal tractory of the straight line, 
whose equation is 



y = a log . + Ja^ 

a + \la^ — x^ 

dy _ Jo' — af 
dx~ X 



THE THEORY OF ROLUNG CURVES. 



15 



de _ Ja?-7* 
dr r* 

= cos"^ 






this is the equation to the orthogonal tractory of a circle whose diameter is 
equal to the constant tangent of the fixed curve, and its constant tangent 
equal to half that of the fixed curve. 

This property of the tractory of the circle may be proved geometrically, 
thus — Let P be the centre of a circle whose radius is PD, and let CD be 
a line constantly equal to the radius. Let BCP be the curve described by 
the point C when the point D is moved along the circumference of the circle, 
then if tangents equal to CD be drawn to the curve, their extremities will 
be in the circle. Let ACH be the curve on which BCP rolls, and let OPE 
be the straight line traced by the pole, let CDE be the common tangent, 
let it cut the circle in D, and the straight line in E. 




Then CD = PD, .'. LDCP^ LDPC, and CP is perpendicular to OE, 
.'. L CPE= LDCP+ LDEP. Take away LDCP-^ L DPC, and there remains 
DPE=DEP, .-. PD=^DE, .-. CE=2PD. 



16 THE THEORY OF ROLLING CURVES. 

Therefore the curve ACH haa a constant tangent equal to the diameter of 
the circle, therefore ACH is the orthogonal tractorj of the straight line, which 
is the tractrix or equitangential curve. 

The operation of finding the fixed curve from the rolled curve is what 
Sir John Leslie calls " divesting a curve of its radiated structure." 

The method of finding the curve which must be rolled on a circle to 
trace a given curve is mentioned here because it generally leads to a double 
result, for the normal to the traced curve cuts the circle in two points, either 
of which may be a point in the rolled curve. 

Thus, if the traced curve be the involute of a circle concentric with the 
given circle, the rolled curve is one of two similar logarithmic spirals. 

If the curve traced be the spiral of Archimedes, the rolled curve may be 
either the hyperbolic spiral or the straight line. 

In the next case, one curve rolls on another and traces a circle. 

Since the curve traced is a circle, the distance between the poles of the 
fixed curve and the rolled curve is always the same; therefore, if we fix the 
rolled curve and roll the fixed curve, the curve traced will still be a circle, 
and, if we fix the poles of both the curves, we may roU them on each other 
without friction. 

Let a be the radius of the traced circle, then the sum or difference of 
the radii of the other curves is equal to a, and the angles which they make 
with the radius at the point of contact are equal, 

.♦. n-=±(a±r,)andn^^ = r,^\ 

dO, _ ±(a±r^ dS, 
drt~ r, dvi' 

If we know the equation between ^j and r,, we may find ^— in terms of r„ 

substitute ± (a ± r,) for r„ multiply by ^ \ and integrate. 

Thus, if the equation between 6^ and r^ be 

r, = a sec $,, 



TEU: THEORY OF ROLLING CURVES. 17 

which is the polar equation of a straight line touching the traced circle whose 
equation is r = ay then 

dd _ a 

dr, ~ r, -Jr.'-a' 
a 



{r,±a)Jr,'±2r,a 
dO^ r^±a a 



dr, r, (r,±a) Jrf±2r^ 



a 



_ 2a _ 2a 

Now, since the rolling curve is a straight line, and the tracing point is 

not in its direction, we may apply to this example the observations which 

have been made upon tractories. 

2a 
Let, therefore, the curve ^ = ^ — 7 be denoted by A, its involute by B, and 

the circle traced by C, then B is the tractory of C; therefore the involute 

2a 
of the curve ^ = ^ — r is the tractory of the circle, the equation of which is 



^ = cos"' /— — I. The curve whose equation is ^'=s — ; seems to be among 

spirals what the catenary is among curves whose equations are between rec- 
tangular co-ordinates ; for, if we represent the vertical direction by the radius 
vector, the tangent of the angle which the curve makes with this line is 
proportional to the length of the curve reckoned from the origin ; the point 
at the distance a from a straight line rolled on this curve generates a circle, 
and when rolled on the catenary produces a straight line ; the involute of this 
curve m the tractory of the circle, and that of the catenary is the tractory 
of the straight line, and the tractory of the circle rolled on that of the straight 
line traces the straight line ; if this curve is rolled on the catenary, it produces 
the straight line touching the catenary at its vertex ; the method of drawing 



18 THE THEORY OF ROLLING CURVES. 

tangents is the same as in the catenary, namely, by describing a circle 
radius is a on the production of the radius vector, and drawing a tangent to the 
circle from the given point. 

In the next case the rolled curve is the same as the fixed curve. It is 
evident that the traced curve wiU be similar to the locus of the intersection 
of the tangent with the perpendicular from the pole ; the magnitude, however, 
of the traced curve will be double that of the other curve; therefore, if we 
call n = <^o^o the equation to the fixed curve, r, = <f>,6, that of the traced curve, 
we have 

also, £^ = f. 

SimUarly, r, = 2p, = 2r,f = A^ Ar, (^J, 0,^6,-2 cos- ^ . 

Similarly, r„ = 2p„., = 2r„_, ^ &c. = 2^ (^^J , 
and ^^f. 

^„ = ^„-7lC0S-f-\ 

'o 

V 
0n = 6. — ncos~^ -^ . 

Let e, become 6^'; 0„ 6,' and ^ , ^. Let ^„^-^„ = a, 

^„^ = ^;-ncos- ^, 
» «. 



a = ^„^- e„ = ^.^-^o-ncos-^ ^' +n cos-^ ^ 

-1 Pn -1 Pn O- , ^0 ~ ^0 

\ cos ^ ^^-^ — COS * -^— = - 4 . 



THE THEORY OF ROLLING CURVES. 19 

Now, cos"^ — is the complement of the angle at which the curve cuts the 

' n 

radius vector, and cos"' — —cos"' -^ is the variation of this angle when 6^ varies 
by an angle equal to a. Let this variation = (^ ; then if 6^ — 6 J = fi, 

^ n n 
Now, if n increases, <f> will diminish ; and if n becomes infinite, 

<^ = ^ + ^ = when a and )8 are finite. 

Therefore, when n is infinite, <}> vanishes ; therefore the curve cuts the radius 
vector at a constant angle ; therefore the curve is the logarithmic spiral. 

Therefore, if any curve be rolled on itself, and the operation repeated an 
infinite number of times, the resulting curve is the logarithmic spiral 

Hence we may find, analytically, the curve which, being rolled on itself, 
traces itself. 

For the curve which has this property, if rolled on itself, and the operation 
repeated an infinite number of times, will still trace itself. 

But, by this proposition, the resulting curve is the logarithmic spiral ; 
therefore the curve required is the logarithmic spiral. As an example of a curve 
rolling on itself, we will take the curve whose equation is 



n=2"a(cos|)". 
-1=2". (sing (oosf-; 
2"a'(cos^")'" 



.'. r^ = 2p,= 2 



r, = 2 



^2-a'(cosg%2-a^(sing (cosg"^'^ 

2"a cos — / n\ „+i 

^^cos-j+(sm-j 



20 THE THEORY OF ROLLING CURVES. 

Now ^1-^0= -cos-^^"= -cos-' cos -" = -^, 

" n+1 
substituting this value of 6^ in the expression for r^, 

r. = 2-'a^cos--J , 

similarly, if the operation be repeated ni times, the resulting curve is 



*afcos— ^^y 
\ n + mj 



When n=l, the curve is 

r = 2a cos 9, 

the equation to a circle, the pole being in the circumference. 

When n = 2, it is the equation to the cardioid 

r = 4a (cos -J . 

In order to obtain the cardioid from the circle, we roll the circle upon 
itself, and thus obtain it by one operation ; but there is an operation which, 
bei6g performed on a circle, and again on the resulting curve, will produce a 
cardioid, and the intermediate curve between the circle and cardioid is 



r = 2 



> / 20\i 



As the operation of rolling a curve on itself is represented by changing n 
into (n + 1) in the equation, so this operation may be represented by changing n 
into (w + i). 

Similarly there may be many other fractional operations performed upon 
the curves comprehended under the equation 

r = 2"a(cos-j. 

We may also find the curve, which, being rolled on itself, will produce a 
given curve, by making 7i= — 1. 



THE THEORY OF ROLLING CURVES. 21 

We may likewise prove by the same method as before, that the result of 
performing this inverse operation an infinite number of times is the logarithmic 
spiral. 

As an example of the inverse method, let the traced line be straight, let 
its equation be 

r<, = 2a sec d^, 
then P^^p,^2a^2a_ 



therefore suppressing the suflSx, 






= ar, 



* • \d0j a ' 



dr 
r 



7i-'' 



■■&-') 



- 2a 
^~l-cos^' 

the polar equation of the parabola whose parameter is 4rt. 

The last case which we shall here consider affords the means of constructing 
two wheels whose centres are fixed, and which shall roll on each other, so that 
the angle described by the first shall be a given function of the angle described 
by the second. 

Let 0^ = (f}0i, then r^ + r^ = a, and -j^ = — ; 

d0^ a-r^' 

Let us take as an example, the pair of wheels which will represent the 
angular motion of a comet in a parabola. 



THE THEORY OF ROLLING CURVES. 



Here 6^ = tan -^ , 



. ^_ 



2 cos' -^ 



a 2 + cos ^1 ' 

therefore the first wheel is an ellipse, whose major axis is equal to | of the 
distance between the centres of the wheels, and in which the distance between 
the foci is half the major axis. 

Now since ^i = 2 tan"' B^ and r^ = a - r„ 

'• 1+ 1 



a ^2(2-^)' 

'-'-±;' 

a 
which is the equation to the wheel which revolves with constant angular velocity. 

Before proceeding to give a list of examples of rolling curves, we shall 
state a theorem which is almost self-evident after what has been shewn pre- 
viously. 

Let there be three curves. A, B, and C. Let the curve A, when rolled 
on itself, produce the curve B, and when rolled on a straight line let it 
produce the curve C, then, if the dimensions of C be doubled, and B be 
rolled on it, it will trace a straight line. 



A Collection of Examples of Rolling Curves. 

First. Examples of a curve rolling on a straight line. 

Ex. 1. When the rolling curve is a circle whose tracing-point is in the 
circumference, the curve traced is a cycloid, and when the point is not in the 
circumference, the cycloid becomes a trochoid. 

Ex. 2. When the rolling curve is the involute of the circle whose radius 
is 2a, the traced curve is a parabola whose parameter is 4a. 



THE THEORY OF ROLLING CURVES. 23 

Ex. 3. When the rolled curve is the parabola whose parameter is 4a, the 
traced curv^e is a catenary whose parameter is a, and whose vertex is distant 
a from the straight line. 

Ex. 4. "When the rolled curve is a logarithmic spiral, the pole traces a 
straight line which cuts the fixed line at the same angle as the spiral cuts 
the radius vector. 

Ex. 5. When the rolled curve is the hyperbolic spiral, the traced curve 
is the tractory of the straight line. 

Ex. 6. When the rolled curve is the polar catenary 



r 2a 



the traced curve is a circle whose radius is a, and which touches the straight 
line. 

Ex. 7. When the equation of the rolled curve is 

the traced curve is the hyperbola whose equation is 

y' = d' + a^. 

Second. In the examples of a straight Hne I'olling on a curve, we shall 
use the letters A^ B, and C to denote the three curves treated of in page 22. 

Ex. 1. When the curve ^ is a circle whose radius is a, then the cui-ve B 
is the involute of that circle, and the curve C is the spiral of Archimedes, r = ad. 

Ex. 2. When the curve ^ is a catenary whose equation is 






the curve B is the tractory of the straight line, whose equation is 

X I 

y = a log , + JcL' — -f^, 

a + V a' - ar" 

and C is a straight line at a distance a from the vertex of the catenary. 



24 THE THEORY OF ROLLING CURVES. 

Ex. 3. When tKe curve A is the polar catenaxy 
the curve B is the tractory of the circle 

and the curve (7 is a circle of which the radius is - . 

Third. Examples of one curve rolling on another, and tracing a straight 
line. 

Ex. 1. The curve whose equation is 

= Ar-"* + &c. + Kr-' + Lr'^ + Jf log r + iVr + &c. + Zt^, 
when rolled on the curve whose equation is 

n — 1 71+ L 

traces the axis of y. 

Ex. 2. The circle whose equation is r = a cos ^ rolled on the circle whose 
radius is a traces a diameter of the circle. 

Ex. 3. The curve whose equation is 



^=J'i- 



1 — versm - , 
a 



rolled on the circle whose radius is a, traces the tangent to the circle. 

Ex. 4. If the fixed curve be a parabola whose parameter is 4a, and if we 
roll on it the spiral of Archimedes r = ad, the pole will trace the axis of the 
parabola. 

Ex. 5. If we roll an equal parabola on it, the focus will trace the directrix 
of the first parabola. 

Ex. 6. If we roll on it the curve ^ = t^ t^® P^^® "^^ ^^^^ ^^® tangent 
at the vertex of the parabola. 



THE THEORY OF ROLLING CURVES. 25 

Ex. 7. If we roll the curve whose equation is 

r = a cos (t^) 
on the ellipse whose equation is 

the pole will trace the axis h. 

Ex. 8. K we roll the curve whose equation ia 

on the hyperbola whose equation is 

the pole will trace the axis h. 

Ex, 9. If we roll the lituus, whose equation is 

on the hyperbola whose equation is 

the pole will trace the asymptote. 

Ex. 10. The cardioid whose equation is 

r = a(H- cos ^), 
rolled on the cycloid whose equation is 

12 = a versin"' - + J2ax - ic*, 
^ a 

traces the base of the cycloid. 

Ex. 11. The curve whose equation is 

= versm-'- + 2^/ 1, 

rolled on the cycloid, traces the tangent at the vertex. 



26 THE THEORY OF ROLLING CURVES. 

Ex. 12. The straight line whose equation is 

r = a sec B, 
rolled on a catenary whose parameter is a, traces a line whose distance from 
the vertex is a. 

Ex. 13. The part of the polar catenary whose equation is 



rolled on the catenary, traces the tangent at the vertex. 

Ex. 14. The other part of the polar catenary whose equation is 

rolled on the catenary, traces a line whose distance from the vertex is equal to 2a. 

Ex. 15. The tractory of the circle whose diameter is a, rolled on the 
tractory of the straight line whose constant tangent is a, produces the straight 
line. 

Ex. 16. The hyperbolic spiral whose equation is 

a 

'■=5' 

rolled on the logarithmic curve whose equation is 

1 ^ 
2/ = alog-, 

traces the axis of y or the asymptote. 

Ex. 17. The involute of the circle whose radius is a, rolled on an orthogonal 
trajectory of the catenary whose equation is 

traces the axis of y. 



Ex. 18. The curve whose equation is 



THE THEORY OF ROLLING CURVES. 27 

rolled on the witch, whose equation is 

traces the asymptote. 

Ex. 19. The curve whose equation is 

r — a tan Q, 
rolled on the curve whose equation is 

traces the axis of y. 

Ex. 20. The curve whose equation is 

2r 



e= 



rolled on the curve whose equation is 

y = / , or r = a tan $, 

traces the axis of y. 

Ex. 21. The curve whose equation is 

r = a (sec d — tan 0), 
rolled on the curve whose equation is 

2/ = alogg+l), 
traces the axis of y. 

Fourth. Examples of pairs of rolling curves which have their poles at a fixed 
distance = a. 



Ce straight line whose equation is ^=sec"'- 
..„ , . 

r 



2a 



The polar catenary whose equation is 0= ±fj I ± 

Ex. 2. Two equal ellipses or hyperbolas centered at the foci. 
Ex. 3. Two equal logarithmic spirals. 

(Circle whose equation is r = 2a cos 6. 

Curve whose equation is ^-/J^ — l + versin"^-. 



Ex. 4. 



28 THE THEORY OF ROLLING CURVES. 

fCaxdioid whose equation is r=2a(l+co8^). 



Ex. 5. 



Ex. 6. 



Ex. 7. 



[Curve whose equation is ^ = sin"*- + log ,— — — . 

(Conchoid, r = a ( secg- 1). 

Icurve, ^ = >A-? 

Spiral of Archimedes, r = a0. 

T T 

Curve, ^ = - + log 



+ sec"^ - 
a 



a ° a 



f Hyperbolic spiral, r=-Q 

Ex. 8. -! 



ICurve, 



a 



e'+l 



1 



Cpse whose equation is ^"^^2+ ~Q' 



Ex. 10. 



(Involute of circle, ^~Ja^^^ ®®^"^ a ' 

'curve, e^J^±2l±log(-±l+J^.±2'^. 



Fifth. Examples of curves rolling on themselves. 
Ex. 1. When the curve which rolls on itself is a circle, equation 

r = a cos 6, 
the traced curve is a cardioid, equation r = a(l+cos^). 
Ex. 2. When it is the curve whose equation is 

r = 2"a (cos-j , 
the equation of the traced curve is 

Ex. 3. When it is the involute of the circle, the traced curve is the spiral 
of Archimedes. 



THE THEORY OF ROLLING CURVES. 29 

Ex. 4. When it is a parabola, the focus traces the directrix, and the vertex 
traces the cissoid. 

Ex. 5. When it is the hyperbolic spiral, the traced curve is the tractory of 
the circle. 

Ex. 6. When it is the polar catenary, the equation of the traced curve is 



J 



2a , . ., r 

1 — versin - . 

r a 



Ex. 7. When it is the curve whose equation is 
the equation of the traced curve is r = a (e' — €~"). 



This paper commenced with an outline of the nature and history of the problem of rolling 
curves, and it was shewn that the subject had been discussed previously, by several geometers, 
amongst whom were De la Hire and Nicolfe in the Memoir es de I'Academie, Euler, Professor 
Willis, in his Principles of Mechanism, and the Rev. H. Holditch in the Cambridge Philosophical 
Transactions. 

None of these authors, however, except the two last, had made any application of their 
methods ; and the principal object of the present communication was to find how far the general 
equations could be simplified in particular cases, and to apply the results to practice. 

Several problems were then worked out, of which some were applicable to the generation 
of curves, and some to wheelwork ; while others were interesting as shewing the relations which 
exist between different curves ; and, finally, a collection of examples was added, as an illus- 
tration of the fertihty of the methods employed. 



[From the Transactions of the Royal Society of Edinburgh, Vol. XX. Part i,] 



III. — On the Equilibrium of Elastic Solids. 



There are few parts of mechanics in which theory has differed more from 
experiment than in the theory of elastic sohds. 

Mathematicians, setting out from very plausible assumptions with respect to 
the constitution of bodies, and the laws of molecular action, came to conclusions 
which were shewn to be erroneous by the observations of experimental philoso- 
phers. The experiments of (Ersted proved to be at variance with the mathe- 
matical theories of Navier, Poisson, and Lame and Clapeyron, and apparently 
deprived this practically important branch of mechanics of all assistance from 
mathematics. 

The assumption on which these theories were founded may be stated thus : — 

Solid bodies are composed of distinct ^molecules, which are kept at a certain 
distance from each other by the opposing principles of attraction and heat. When 
the distance between two molecules is changed, they act on each other with a force 
whose direction is in the line joining the centres of the molecules, and whose 
magnitude is equal to the change of distance multiplied into a function of the 
distance which vanishes when that distance becomes sensible. 

The equations of elasticity deduced from this assumption contain only one 
coefficient, which varies with the nature of the substance. 

The insufficiency of one coefficient may be proved from the existence of 
bodies of different degrees of solidity. 

No effort is required to retain a liquid in any form, if its volume remain 
unchanged; but when the form of a solid is changed, a force is called into 
action which tends to restore its former figure ; and this constitutes the differ- 



THE EQUILIBRITJM OF ELASTIC SOLIDS. 31 

ence between elastic solids and fluids. Both tend to recover their vohirne, but 
fluids do not tend to recover their shape. 

Now, since there are in nature bodies which are in every intermediate state 
from perfect soHdity to perfect liquidity, these two elastic powers cannot exist 
in every body in the same proportion, and therefore all theories which assign to 
them an invariable ratio must be erroneous. 

I have therefore substituted for the assumption of Navier the following 
axioms as the results of experiments. 

If three pressures in three rectangular axes be applied at a point in an 
elastic solid, — 

1. TTie sum of the three pressures is proportional to the sum of the com- 
pressions ichich they produce. 

2. The difference between two of the pressures is propo7'tional to the differ- 
ence of the compressions which they produce. 

The equations deduced from these axioms contain two coefficients, and differ 
from those of Navier only in not assuming any invariable ratio between the 
cubical and linear elasticity. They are the same as those obtained by Professor 
Stokes from his equations of fluid motion, and they agree with all the laws of 
elasticity which have been deduced from experiments. 

In this paper pressures are expressed by the number of units of weight to 
the unit of surface ; if in English measure, in pounds to the square inch, or 
in atmospheres of 15 pounds to the square inch. 

Compression is the proportional change of any dimension of the solid caused 
by pressure, and is expressed by the quotient of the change of dimension divided 
by the dimension compressed'". 

Pressure will be understood to include tension, and compression dilatation ; 
pressure and compression being reckoned positive. 

Elasticity is the force which opposes pressure, and the equations of elasticity 
are those which express the relation of pressure to compression f. 

Of those who have treated of elastic solids, some have confined themselves 
to the investigation of the laws of the bending and twisting of rods, without 

* The laws of pressure and compression may be found in the Memoir of Lam6 and Clapeyrou. St^t- 
note A. 

t See note B. 



32 THE EQUIUBRIUM OF ELASTIC SOLIDS. 

considering the relation of the coefficients which occur in these two cases; 
while others have treated of the general problem of a solid body exposed to 
any forces. 

The investigations of Leibnitz, Bernoulli, Euler, Varignon, Young, La Hire, 
and Lagrange, are confined to the equilibrium of bent rods; but those of 
Navier, Poisson, Lam^ and Clapeyron, Cauchy, Stokes, and Wertheim, are 
principally directed to the formation and application of the general equations. 

The investigations of Navier are contained in the seventh volume of the 
Memoirs of the Institute, page 373; and in the AnnoUes de Chimie et de 
Physique, 2^ Sdrie, xv. 264, and xxxviii. 435 ; L'AppUcati(m de la Micanique, 
Tom. I. 

Those of Poisson in Mem. de I'lnstitut, vm. 429 ; Annales de Chimie, 2" 
S^rie, XXXVI, 334 ; xxxvii. 337 ; xxxvtil 338 ; xlu. Journal de VEcole 
Polytechnique, cahier xx., with an abstract in Annales de Chimie for 1829. 

The memoir of MM. Lam^ and Clapeyron is contained in Crelle's Mathe- 
matical Journal, Vol. vii. ; and some observations on elasticity are to be found 
in Lamp's Cours de Physique, 

M. Cauchy's investigations are contained in his Exercices d! Analyse, Vol. in. 
p. 180, published in 1828. 

Instead of supposing each pressure proportional to the linear compression 
which it produces, he supposes it to consist of two parts, one of which is pro- 
portional to the linear compression in the direction of the pressure, while the 
other is proportional to the diminution of volume. As this hypothesis admits 
two coefficients, it differs from that of this paper only in the values of the 
coefficients selected. They are denoted by K and h, and K^fi — ^m, k = m. 

The theory of Professor Stokes is contained in Vol. vin. Part 3, of the 
Cambridge Philosophical Transactions, and was read April 14, 1845. 

He states his general principles thus : — " The capability which solids possess 
of being put into a state of isochronous vibration, shews that the pressures 
called into action by small displacements depend on homogeneous functions of 
those displacements of one dimension. I shall suppose, moreover, according to 
the general principle of the superposition of small quantities, that the pressures 
due to different displacements are superimposed, and, consequently, that the 
pressures are linear functions of the displacements." 



THE EQUILIBRIUM OF ELASTIC SOLIDS. 33 

Having assumed the proportionality of pressure to compression, he proceeds 
to define his coefficients.— "Let -^8 be the pressures corresponding to a uniform 
linear dilatation 8 when the solid is in equilibrium, and suppose that it becomes 
mA8, in consequence of the heat developed when the solid is in a state of rapid 
vibration. Suppose, also, that a displacement of shifting parallel to the plane 
xy, for which 8x = kx, Sy= - hj, and hz = 0, calls into action a pressure - Bk 
on a plane perpendicular to the axis of x, and a pressure Bk on a plane 
perpendicular to the axis of y; the pressure on these planes being equal and 
of contrary signs; that on a plane perpendicular to z being zero, and the tan- 
gential forces on those planes being zero." The coefficients A and B, thus 

defined, when expressed as in this paper, are ^ = 3/x,, B = -. 

Professor Stokes does not enter into the solution of his equations, but gives 
their results in some particular cases. 

1. A body exposed to a uniform pressure on its whole surface. 

2. A rod extended in the direction of its length. 

3. A cylinder twisted by a statical couple. 

He then points out the method of finding A and B from the last two cases. 

While explaining why the equations of motion of the luminiferous ether are 
the same as those of incompressible elastic solids, he has mentioned the property 
of jylasticity or the tendency which a constrained body has to relieve itself 
from a state of constraint, by its molecules assuming new positions of equi- 
librium. This property is opposed to Hnear elasticity ; and these two properties 
exist in all bodies, but in variable ratio. 

M. Wertheim, in Annales de Chimie, 3« Sdrie, xxiii., has given the results 
of some experiments on caoutchouc, from which he finds that K=k, or fi = ^m; 
and concludes that k = K in all substances. In his equations, fi is therefore 
made equal to f m. 

The accounts of experimental researches on the values of the coefficients 
are so numerous that I can mention only a few. 

Canton, Perkins, (Ersted. Aime, CoUadon and Sturm, and Regnault, have 
determined the cubical compressibilities of substances; Coulomb, Duleau, and 
Giulio, have calculated the linear elasticity from the torsion of wires; and a 
great many observations have been made on the elongation and bending of beams. 

VOL. I. ^ 



34 THE EQUILIBRIUM OF ELASTIC SOLIDS. 

I have found no account of any experiments on the relation between the 
doubly refracting power communicated to glass and other elastic solids by com- 
pression, and the pressure which produces it^^" ; but the phenomena of bent glass 
seem to prove, that, in homogeneous singly-refracting substances exposed to 
pressures, the principal axes of pressure coincide with the principal axes of 
double refraction ; and that the diflference of pressures in any two axes is 
proportional to the difference of the velocities of the oppositely polarised rays 
whose directions are parallel to the third axis. On this principle I have 
calculated the phenomena seen by polarised light in the cases where the solid 
is bounded by parallel planes. 

In the following pages I have endeavoured to apply a theory identical 
with that of Stokes to the solution of problems which have been selected on 
account of the possibility of fulfilling the conditions. I have not attempted to 
extend the theory to the case of imperfectly elastic bodies, or to the laws of 
permanent bending and breaking. The solids here considered are supposed not 
to be compressed beyond the limits of perfect elasticity. 

The equations employed in the transformation of co-ordinates may be found 
in Gregory's Solid Geometry. 

I have denoted the displacements by Zx, By, Bz. They are generally denoted 
by a, /8, y ; but as I had employed these letters to denote the principal axes 
at any point, and as this had been done throughout the paper, I did not alter 
a notation which to me appears natural and intelligible. 



The laws of elasticity express the relation between the changes of the 
dimensions of a body and the forces which produce them. 

These forces are called Pressures, and their effects Compressions. Pressures 
are estimated in pounds on the square inch, and compressions in fractions of the 
dimensions compressed. 

Let the position of material points in space be expressed by their co-ordinates 
X, y, and z, then any change in a system of such points is expressed by giving 
to these co-ordinates the variations Bx, By, Bz, these variations being functions of 

X, y, 2. 

* See note C. 



THE EQUILIBRIUM OF ELASTIC SOLIDS. 35 

The quantities Sx, Sy, 8z, represent the absolute motion of each point in 
the directions of the three co-ordinates ; but as compression depends not on 
absolute, but on relative displacement, we have to consider only the nine 
quantities — 



dSx 


dSx 


dhx 


dx ' 


dy' 


dz' 


dSy 
dx ' 


dhy 
dy' 


dSij 

dz ' 


dSz 
dx' 


dhz 

dy- 


dBz 

dz ' 



Since the number of these quantities is nine, if nine other independent 
quantities of the same kind can be found, the one set may be found in terms 
of the other. The quantities which we shall assume for this purpose are— 

1. Three compressions, — , —■ , — , in the directions of three principal 

a Id y 

axes a, yS, y. 

2. The nine direction-codnes of these axes, with the six connecting equa- 
tions, leaving three independent quantities. (See Gregory's Solid Geometry.) 

3. The small angles of rotation of this system of axes about the axes of 
x, y, z. 

The cosines of the angles which the axes of x, y, z make with those of 
a, ^, y are 

cos(aOa-)=aj, cos {^Ox) = \, co%(yQ)x) = c,, 
cos (aOy) = tto, _cos {fiOy) = h„, cos (yO^/) = c., 
cos (aOz) =a3, cos (/SOz) =63, cos {yOz) = c,. 

These direction-cosines are connected by the six equations, 
a^ + h{ + Ci' = 1 , «i«s + ^h + CjC, = 0, 

a./ -I- h^ + c,' = 1 , a^a^ + h.h^ + cx^ = 0, 

a; + 63' + Gj' = 1 , a/t, + bj), + c^c, = 0. 

The rotation of the system of axes a, 13, y, round the axis of 
x, from y to z, =B0^, 
y, from z to x, =S^j, 
z, from x to y, =^0/, 



36 



THE EQUILIBRIUM OF ELASTIC SOLIDS. 



By resolving the displacements 8a, h/S, By, B6„ B9.„ Z6„ in the directions 
of the axes x, y, z, the displacements in these axes are found to be 
hx = a,8a + h,Bp + c3y -Be^ + Bd,y, 
By = aM + h,Bl3 -f c,By - Bd,x + Bd.z, 
Bz = a,Ba + hM + CsBy - BO^ + Bd,x. 
Sa .^ ^Si8 



But 



B^^rf, and 8y = y^, 



and Q. = a^x + a^ + a.^, /3 = b,x + h^ + h.^, and y = c,x + c,y -h c^z. 

Substituting these values of Sa, Sy8, and By in the expressions for Bx, By, 
Bz, and differentiating with respect to x, y, and z, in each equation, we obtain 
the equations 

dBx Ba, ,. 8/8,2 , ^y 



dy a ^ y 

dBz _ Ba 
dz 



a p y 



(1)- 



dBx Ba B^ T J By ,5s/, 

dy a ' ^ y 



a 

dBx Ba 
dz a 

Ba 



BI3 

J' 

8^ 



dz a p y 



dBy Ba BB T ^ By 

dx a p y 



Be, 

c.f^ + Bdi 

Be, 



-J— = — ctjCti + -^ 6361 + -^ C3C1 + 8^2 



dZz 
dx 

dBz 



8^ 
a 

Sa 



8^ 
S/8 



r 






Be, 



Equations of 
compression. 



{2). 



Equations of the equilibnum of an element of the solid. 
The forces which may act on a particle of the solid are : — 

1. Three attractions in the direction of the axes, represented by X, Y, Z. 

2. Six pressures on the six faces. 



THE EQUILIBRIUM OF ELASTIC SOLIDS. 



37 



3. Two tangential actions on each face. 

Let the six faces of the small parallelopiped be denoted by x^, 3/,, z„ x^ y„ 
and z,, then the forces acting on x^ are : — 

1. A normal pressure jp, acting in the direction of x on the area dydz, 

2. A tangential force g, acting in the direction of y on the same area. 

3. A tangential force q^ acting in the direction of z on the same area, 
and so on for the other five faces, thus : — 

Forces which act in the direction of the axes of 

a; 2/ z 



On the face a:, 


— 'p^dydz 


- q^dydz 


-q.'dydz 


^. 


{P^'r J^dx)dydz 


(^3 + 7^ ^^) c?yc?x 


(q.'+-^^dx)dydz 


2/1 


— q^dzdx 


—p^dzdx 


— q.dzdx 


y-x 


{q\ + ^dy)dzdx 


{p.+^dy)dzdx 


(q, + ^dy)dzdx 


Zi 


— q^dxdy 


— q^dxdy 


—p^dxdy 


^2 


fe+ -4^dz)dxdy 


(q^+^dz)dxdy 


(p. + ^dz)dxdy 


Attractions, 


pXdxdydz 


p Ydxdydz 


pZdxdydz 



Taking the moments of these forces round the axes of the particle, we find 

?i' = ?i, q^=q.^ qz=qz', 

and then equating the forces in the directions of the three axes, and dividing 
by dx, dy, dz, we find the equations of pressures, 



dy dz dx ^ 
dz dx dy '^ 



Equations of Pressures. 



(3). 



38 



THE EQUILIBRIUM OF ELASTIC SOLIDS. 



The resistance which the sohd opposes to these pressures is called Elasticity, 
and is of two kinds, for it opposes either change of volume or change of Jigure. 
These two kinds of elasticity have no necessary connection, for they are possessed 
in very different ratios by different substances. Thus jelly has a cubical elas- 
ticity little different from that of water, and a linear elasticity as small as we 
please ; while cork, whose cubical elasticity is very small, has a much greater 
Imear elasticity than jelly. 

Hooke discovered that the elastic forces are proportional to the changes 
that excite them, or as he expressed it, " Ut tensio sic vLs." 

To fix our ideas, let us suppose the compressed body to be a parallelepiped, 
and let pressures Pi, Pj, P3 act on its faces in the direction of the axes 
a> A y, which will become the principal axes of compression, and the com- 



pressions will be 



So. 8^ Sy 
a' ^' y 



The fundamental assumption from which the following equations are deduced 
is an extension of Hooke's law, and consists of two parts. 

I. The sum of the compressions is proportional to the sum of the pressures. 

II. The difference of the compressions is proportional to the difference of 
the pressures. 



These laws are expressed by the following equations 



I. (P. + P, + P.) = 3,(^ + f + ^ 



(4). 



II. 



(P,-P,) = m 



(P._p.) = „,g_^ 



(P.-P,) = m 



rv ^rts T Equations of Elasticity. 

h 
7 

By Ba 



(5). 



The quantity fj. is the coefiicient of cubical elasticity, and m that of linear 
elasticity. 



THE EQUILrBRIUM OF ELASTIC SOLmS. 



39 



By solving these equations, the values of the pressures P„ P,, P„ and the 

8a 8^ Sy , r J 

compressions — ' ~S ' ^^7 ^^ found. 



a \9/x 3m/ ^ ^ m 

! = (!_ M(p. + P, + p.) + lp, 

j3 \9/x 3 m/ ^ * ^ ?7i ' 

?r = (_L_ i\(P_+P_+P_) + ip_ 

y \9/z 3m/ ^ ^ m 



(6). 



(7). 



From these values of the pressures in the axes a, )8, y, may be obtained.. 
the equations for the axes x, y, z, by resolutions of pressures and compressions*. 



For 



and 



q = aaP^ + hhP, + ccP, ; 
, . . IdZx , d%y , d8z\ . d8x' 

, . V IdZx . d8y , d8z\ dBy 

, , , fdSx , d8y , rfSj\ , dSz 

m /c?Sz c?Sx 



(8)- 



2 Vo?a; c?2 






.(9). 



See the Memoir of Lame and Clapeyron, and note A. 



40 



THE EQUIUBRIUM OP ELASTIC SOLIDS. 



d$X /I 1 \ , , , N , 1 



(10). 



dy * ax ' m^ 
dz dy m ^ 



d^ 
dx 



dz m^ 



(11). 



By substituting in Equations (3) the values of the forces given in Equa- 
tions (8) and (9), they become 



(12). 



These are the general equations of elasticity, and are identical with those 
of M. Cauchy, in his Exercices d' Analyse, Vol. ni., p. 180, published in 1828, 

where h stands for m, and K for ft - o" > and those of Mr Stokes, given in the 

Cambridge Philosophical Transactions, Vol. viii., part 3, and numbered (30); 

in his equations ^ = 3/x, B = — . 

If the temperature is variable from one part to another of the elastic 

soHd, the compressions -y- , -r^, -J^ , at any point will be diminished by a 

quantity proportional to the temperature at that point. This prmciple is applied 
in Cases X. and XI. Equations (10) then become 



THE EQUILIBRIUM OF ELASTIC SOLIDS. 



41 






dy 



^ = fe - 3mj (P^-^P^+P^) + '^^^^P^ 



(13). 



CfV being the linear expansion for the temperature v. 

Having found the general equations of the equilibrium of elastic solids, I 
proceed to work some examples of their application, which afford the means of 
determining the coefficients /t, m, and o), and of calculating the stiffness of 
solid figures. I begin with those cases in which the elastic soHd is a hollow 
cylinder exposed to given forces on the two concentric cylindric surfaces, and 
the two parallel terminating planes. 

In these cases the co-ordinates x, y, z are replaced by the co-ordinates 
x = x, measured along the axis of the cylinder. 
2/ = r, the radius of any point, or the distance from the axis. 
z — rd, the arc of a circle measured from a fixed plane passing 
through the axis. 

Px = o, are the compression and pressure in the direction of the 
axis at any point. 

-^ = -J— , Pi =p, are the compression and pressure in the direction of the 

radius. 

dBz dhrd Br . . _ . , ,. . - 1 

~dz~'db¥~l^' JP8 = ?, are the compression and pressure m the direction of the 

tangent. 

Equations (9) become, when expressed in terms of these co-ordinates — 

m doO 



dZx 
dx 



dSx 
dx 



m dB0 

m dSx 
dr 



*=2 



.(14). 



The length of the cylinder is h, and the two radii a, and a, in every 
VOL. I. G 



42 THE EQUIUBRnJM OF ELASTIC SOLIDS. 



Case I. 



The first equation is applicable to the case of a hollow cylinder, of which 
the outer surface is fixed, while the inner surface is made to turn through 
a small angle Bd, by a couple whose moment is M. 

The twisting force M is resisted only by the elasticity of the solid, and 
therefore the whole resistance, in every concentric cylindric surface, must be equal 
to M. 

The resistance at any point, multiplied into the radius at which it acts, is 
expressed by 

m „ dhd 

Therefore for the whole cylindric surface 

ar 
Whence 8,=_^^ (1,_1.) , 

^^ "' = 2^&-i) ('«>■ 

The optical effect of the pressure of any point is expressed by 

I=<oq,b = <o.^^ (15). 

Therefore, if the solid be viewed by polarized light (transmitted parallel to 
the axis), the difference of retardation of the oppositely polarized rays at any 
point in the solid will be inversely proportional to the square of the distance fi-om 
the axis of the cylinder, and the planes of polarization of these lays will be 
inclined 45" to the radius at that point. 

The general appearance is therefore a system of coloured rings arranged 
oppositely to the rings in uniaxal crystals, the tints ascending in the scale as 
they approach the centre, and the distance between the rings decreasing towards 
the centre. The whole system is crossed by two dark bands inclined 45* to the 
plane of primitive polarization, when the plane of the analysing plate is perpen- 
dicular to that of the first polarizing plate. 



THE EQUILIBRIUM OF ELASTIC SOLIDS. 43 

A jelly of isinglass poured when hot between two concentric cylinders forms, 
when cold, a convenient solid for this experiment ; and the diameters of the rings 
may be varied at pleasure by changing the force of torsion appUed to the interior 
cylinder. 

By continuing the force of torsion while the jeUy is allowed to dry, a hard 
plate of isinglass is obtained, which still acts in the same way on polarized light, 
even when the force of torsion is removed. 

It seems that this action cannot be accounted for by supposing the interior 
parts kept in a state of constraint by the exterior parts, as in, unannealed and 
heated gla^s ; for the optical properties of the plate of isinglass are such as 
would indicate a strain preserving in every part of the plate the direction of 
the original strain, so that the strain on one part of the plate cannot be main- 
tained by an opposite strain on another part. 

Two other uncrystallised substances have the power of retaining the polariz- 
ing structure developed by compression. The first is a mixture of wax and resin 
pressed into a thin plate between two plates of glass, as described by Sir David 
Brewster, in the Philosophical TransoLctions for 1815 and 1830. 

When a compressed plate of this substance is examined with polarized light, 
it is observed to have no action on light at a perpendicular incidence ; but when 
inclined, it shews the segments of coloured rings. This property does not belong 
to the plate as a whole, but is possessed by every part of it. It is therefore 
similar to a plate cut from a uniaxal crystal perpendicular to the axis. 

I find that its action on light is like that of a jpositive crystal, while that 
of a plate of isinglass similarly treated would be negative. 

The other substance which possesses similar properties is gutta percha. This 
substance in its ordinary state, when cold, is not transparent even in thin films; 
but if a thin film be drawn out gradually, it may be extended to more than 
double its length. It then possesses a powerful double refraction, which it 
retains so strongly that it has been used for polarizing light""'. As one of its 
refractive indices is nearly the same as that of Canada balsam, while the other 
is very different, the common surface of the gutta percha and Canada balsam 
will transmit one set of rays much more readdy than the other, so that a film 
of extended gutta percha placed between two layers of Canada balsam acts like 

* By Dr Wright, I believe. 



44 THE EQUILIBRIUM OF ELASTIC SOLIDS. 

a plate of nitre treated in the same way. That these films are in a state of 
constraint may be proved by heating them slightly, when they recover their 
original dimensions. 

As all these permanently compressed substances have passed their limit of 
perfect elasticity, they do not belong to the class of elastic solids treated of in 
this paper ; and as I cannot explain the method by which an imcrystallised body 
maintains itself in a state of constraint, I go on to the next case of twisting, 
which has more practical importance than any other. This is the case of a 
cylinder fixed at one end, and twisted at the other by a couple whose moment 
is M. 

Case II. 

In this case let hB be the angle of torsion at any point, then the resistance 
to torsion in any circular section of the cylinder is equal to the twisting force M, 

The resistance at any point in the circular section is given by the second 
Equation of (14). 



?2 = 1^^ 



dx ' 



This force acts at the distance r from the axis ; therefore its resistance to torsion 
will be q.r, and the resistance in a circular annulus will be 

q^r^Ttrdr = mirr' -r- dr 

and the whole resistance for the hollow cylinder will be expressed by 

„, mn dS6 , ^ ,. /,^v 






720 M 



^(-1-] (17). 



In this equation, m is the coefl&cient of linear elasticity; a^ and a^ are the 
radii of the exterior and interior surfaces of the hollow cyUnder in inches ; M is 
the moment of torsion produced by a weight acting on a lever, and is expressed 



THE EQUILIBRIUM OF ELASTIC SOLIDS. 45 

bj the product of the number of pounds in the weight into the number of inches 
in the lever; b is the distance of two points on the cylinder whose angular 
motion is measured by means of indices, or more accurately by small mirrors 
attached to the cylinder ; n is the difference of the angle of rotation of the two 
indices in degrees. 

This is the most accurate method for the determination of m independently 
of /x, and it seems to answer best with thick cylinders which cannot be used 
with the balance of torsion, as the oscillations are too short, and produce a 
vibration of the whole apparatus. 



Case III. 

A hollow cylinder exposed to normal pressures only. When the pressures 
parallel to the axis, radius, and tangent are substituted for p^, p^, and pt, 
Equations (10) become 

S = (i-34)(^+^-^^) + ^ (^«)- 

^^t^(±-±]io+p + q) + :^q (20). 

By multiplying Equation (20) by r, differentiating with respect to r, and 

comparing this value of —j— with that of Equation (19), 



p-q _(J__ _1\ /^ . ^ . ^\ _ i ^ 
rm " \9/x 3m/ \dr dr drj m dr ' 



The equation of the equilibrium of an element of the solid is obtained by 
considering the forces which act on it in the direction of the radius. By 
equating the forces which press it outwards with those pressing it rnwarde, we 
find the equation of the equiHbrium of the element, 

ir£ = 4 (21). 

r dr 



46 THE EQUILIBRIUM OF ELASTIC SOLIDS. 

By comparing this equation witli the last, we find 

\9fi Zmj dr \9/i ^ 3m/ \dr ^ drj 
Integrating, 

Since o, the longitudinal pressure, is supposed constant, we may assume 

c -(^-^]o 
' \9u, 3m/ . , . 

c. = 12 =(^ + g)- 

9/x, 3 m 
Therefore q—p = c^ — 2p, therefore by (21), 

a linear equation, which gives 






1 ^c, 
^ = ^3^ + 2- 

The coefficients Cj and Cj must be found from the conditions of the surface 
of the soHd. If the pressure on the exterior cylindric surface whose radius is a, 
be denoted by A,, and that on the interior surface whose radius is a^ by A,, 

then p = h^ when r = ai 
and p = h.j when r = a^ 
and the general value of p is 

_a^h^ — a^\ a^a^ h^ — h^ /22\ 

^" a,' -a,' ^ oT^^ ^ ^' 






2-i'=2i^ ^73^- ''y (21). 



*= «.'-«.' +^^57::^' (^^^■ 

/=5<.(^-2)=-26<.^"A^. (24). 

This last equation gives the optical eflfect of the pressure at any point. The 
law of the magnitude of this quantity is the inverse square of the radius, as in 



THE EQUILIBRIUM OF ELASTIC SOLIDS. 47 

Case I. ; but the direction of the principal axes ia different, as in this case they 
are parallel and perpendicular to the radius. The dark bands seen by polarized 
Ught wiU therefore be parallel and perpendicular to the plane of polarisation, in- 
stead of being inclined at an angle of 45", as in Case I. 

By substituting in Equations (18) and (20), the values of p and q given in 
(22) and (23), we find that when r = a,. 



hx (l\( ^aX-ct'h-X . 2 / a,%-a,%\ ] 



X \9/x 



= o(^ + ~] + 2{Ka,^-Ka,^) 



1/1 1 



.(25). 



,9/x 3m/ ' ^ ' ' ' 'Ui,'-a,'\9fj, 3mJ 
r 9/x \ a/ — a/ / 3? 



When r = a., - ^ ^ fo4-2 ^4-^) + ^^^ ( - ^._^. ' ' -o 



(26). 



~ VSft 3my "^ ' a; - a,' \ 9/x ^ 3m / ^ cv - a,' 1,9/x "^ 3m/ J 

From these equations it appears that the longitudinal compression of cylin- 
dric tubes is proportional to the longitudinal pressure referred to unit of surface 
when the lateral pressures are constant, so that for a given pressure the com- 
pression is inversely as the sectional area of the tube. 

These equations may be simplified in the following cases : — 

1. When the external and internal pressures are equal, or h^ = h^. 

2. When the external pressure is to the internal pressure as the square of 
tlie interior diameter is to that of the exterior diameter, or when a^-h^ = a^-h^. 

3. When the cylinder is soHd, or when a. = 0. 

4. When the solid becomes an indefinitely extended plate with a cylindric 
hole in it, or when a^ becomes infinite. 

5. When pressure is applied only at the plane surfaces of the solid cylinder, 
and the cylindric surface is prevented from expanding by being inclosed in a 

strong case, or when — = 0. 

6. When pressure is applied to the cylindric surface, and the ends are 
retained at an invariable distance, or when — = 0. 

X 



48 



THE EQUILIBRIUM OF ELASTIC SOLIDS. 



1. When ^ji = A„ the equations of compression become 



\9fi'*"3mj"'"^ '\9ij. 3m 



(27). 



7 = i('>+2^) + 3i(^-<') 

When hi = hi = o, then 

Zx _hr _ \ 
X ~ r " Sfi' 

The compression of a cylindrical vessel exposed on all sides to the same 
hydrostatic pressure is therefore independent of m, and it may be shewn that 
the same is true for a vessel of any shape. 



2. When a,% = a^% 



^ \9yx "^ 3m/ 



Bx 

X 



7 = |w + 3l(3^--»)^ 



(28). 



In this case, when o = 0, the compressions are independent of /x. 
3. In a solid cylinder, aj = 0, 

The expressions for — and — are the same as those in the first case, when 
h^ — hf 

When the lon^tudinal pressure o vanishes, 

Bx 

X 



r ' \9/x 3m/ ' 



THE EQUILIBRIUM OF ELASTIC SOLIDS. 



49 



When the cylinder ia pressed on the plane sides only, 

8x 



r \9fi dmj 



4. When the solid is infinite, or when a, is infinite, 



p = K--._a-(\-K) 



I=<o{p-q)=-^a.;{h,-h,) 
r 9/x ^ ' 3m ^ ' 



(29). 



5. When 8r = in a solid cylinder, 

Zx Zo 



6. When 



X 2m + 3/A 

So; _ hr _ 2>h 
x~ * r ~ m + 6iM 



.(30). 



Since the expression for the efiect of a longitudinal strain is 
Bx 



if we make 

VOL. I. 



-=o(— + —) 
X \9/i, 3m/ ' 



r, 9mu, ^, 8x 1 

E = ^ , then — = o ^^ 

m + 6/x cc E 



(31). 



50 THE EQUILIBRIUM OF ELASTIC SOLIDS. 

The quantity E may be deduced from experiment on the extension of wires 
or rods of the substance, and /x is given in terms of m and E by the equation, 

„ = _^!!L_ (32), 

^^^ ^ = S (^^)' 

P being the extending force, h the length of the rod, s the sectional area, 
and Bx the elongation, which may be determined by the deflection of a wire, 
as in the apparatus of S' Gravesande, or by direct measurement. 



Case IV. 

The only known direct method of finding the compressibihty of liquids is 
that employed by Canton, (Ersted, Perkins, Aime, &c. 

The liquid is confined in a vessel with a narrow neck, then pressure is 
applied, and the descent of the liquid in the tube is observed, so that the 
difference between the change of volume of liquid and the change of internal 
capacity of the vessel may be determined. 

Now, since the substance of which the vessel is formed is compressible, a 
change of the internal capacity is possible. If the pressure be applied only to 
the contained liquid, it is evident that the vessel will be distended, and the 
compressibihty of the liquid will appear too great. The pressure, therefore, is 
commonly applied externally and internally at the same time, by means of a 
hydrostatic pressure produced by water compressed either in a strong vessel or 
in the depths of the sea. 

As it does not necessarily follow, from the equality of the external and 
internal pressures, that the capacity does not change, the equilibrium of the 
vessel must be determined theoretically. (Ersted, therefore, obtained from Poisson 
his solution of the problem, and applied it to the case of a vessel of lead. 
To find the cubical elasticity of lead, he appUed the theory of Poisson to the 
numerical results of Tredgold. As the compressibility of lead thus found was 
greater than that of water, (Ersted expected that the apparent compressibility 
of water in a lead vessel would be negative. On making the experiment the 
apparent compressibihty was greater in lead than in glass. The quantity found 



THE EQUILIBRrcrM OF ELASTIC SOLIDS. 51 

by Tredgold from the extension of rods was that denoted by E, and the value 
of ft deduced from E alone by the formulae of Poisson cannot be true, unless 

— = |-; and as — for lead is probably more than 3, the calculated compressi- 
bility is much too great. 

A similar experiment was made by Professor Forbes, who used a vessel of 
caoutchouc. As in this case the apparent compressibility vanishes, it appears 
that the cubical compressibihty of caoutchouc is equal to that of water. 

Some who reject the mathematical theories as unsatisfactory, have conjec- 
tured that if the sides of the vessel be sufficiently thin, the pressure on both 
sides being equal, the compressibility of the vessel will not affect the result. 
The following calculations shew that the apparent compressibility of the liquid 
depends on the compressibility of the vessel, and is independent of the thickness 
when the pressures are equal. 

A hollow sphere, whose external and internal radii are a^ and a,, is acted 
on by external and internal normal pressures h^ and K, it is required to deter- 
mine the equilibrium of the elastic solid. 

The pressures at any point in the solid are : — 

1. A pressure p in the direction of the radius. 

2. A pressure q in the perpendicular plane. 

These pressures depend on the distance from the centre, which is denoted 
by r. 

The compressions at any point are -.— in the radial direction, and — in 
the tangent plane, the values of these compressions are : — 

fr=[h-^^P^''i)*h^ ('")• 

T = fe-3fJ(^ + 2,) + l5 (35). 

Multiplying the last equation by r, differentiating with respect to r, and 
equating the result with that of the first equation, we find 



52 



THE EQUILIBRITTM OF ELASTIC SOLIDS. 



Since the forces whicli act on the particle in the direction of the radius 
must balance one another, or 



2qdrde +p (rdey =(^p + ^d7^(r + dry 6, 



_r dp 



therefore ^""-^ = 2 37 ^^^^' 

Substituting this value of q -p in the preceding equation, and reducing, 

therefore 



^ + 2^ = 0. 
dr dr 



Integrating, 
But 
and the equation becomes 

therefore 



p-\-2q = c,. 
r dp , 



dp 
dr 



+ 3^-^-i = 0, 



1 c. 



Since p = h, when r = a.,, and p = K when r = a,, the value of p at any 
distance is found to be 



^~ a^-af r' a^-a,' 
9- a,'-ai "^^ 7^ <-a/ 



(37). 
.(38). 



When r = a„ -y = -^r:^^ - + t ^^ ^^737^3 ^ 

~ a,' - a/ U 2»i/ a/ - «/ \jx 2wi/ _ 
When the external and internal pressures are equal 



.(39). 



h^ = h.,=p = q, and -y- 



SV K 



.(40), 



THE EQUILIBRIUM OF ELASTIC SOLIDS. 53 

the change of internal capacity depends entirely on the cubical elasticity of the 
vessel, and not on its thickness or linear elasticity. 

When the external and internal pressures are inversely as the cubes of the 
radii of the surfaces on which they act, 



aX = a,%, p = ^ K q= -i^K 



when r = r- — ^ ' 



(41). 



V 2 ^^ 

In this case the change of capacity depends on the linear elasticity alone. 

M. Regnault, in his researches on the theory of the steam engine, has 
given an account of the experiments which he made in order to determine 
with accuracy the compressibility of mercury. 

He considers the mathematical formulae very uncertain, because the theories 
of molecular forces from which they are deduced are probably far from the 
truth ; and even were the equations free from error, there would be much 
uncertainty in the ordinary method by measuring the elongation of a rod of 
the substance, for it is diflScult to ensure that the material of the rod is the 
same as that of the hollow sphere. 

He has, .therefore, availed himself of the results of M. Lam6 for a hollow 
sphere in three different cases, in the first of which the pressure acts on the 
interior and exterior surface at the same time, while in the other two cases 
the pressure is applied to the exterior or interior surface alone. Equation (39) 
becomes in these cases, — 

1. When ^1 = /ij, -^ = — and the compressibility of the enclosed liquid being 
/x,, and the apparent diminution of volume S'F, 

v-.£-;) «■ 

2. When /i, = 0, 



54 THE EQUILIBRIUM OF ELASTIC SOLIDS. 

3. When h,^0, 

8V_ h K , 9^\ 

V a^-a^ \ii ^ m ^ ' V2 J 

M. Lamp's equations differ from these only in assuming that fi, = |-m. If 
this assumption be correct, then the coefficients /u,, m, and jMj, may be found 
from two of these equations ; but since one of these equations may be derived 
from the other two, the three coefficients cannot be found when /u, is supposed 
independent of m. In Equations (39), the quantities which may be varied at 
pleasure are \ and h^, and the quantities which may be deduced from the 
apparent compressions are, 



'■=G+4)^°<^S-i)=^" 



therefore some independent equation between these quantities must be found, 
and this cannot be done by means of the sphere alone; some other experiment 
must be made on the liquid, or on another portion of the substance of which 
the vessel is made. 

The value of /x^, the elasticity of the liquid, may be previously known. 

The linear elasticity m of the vessel may be found by twisting a rod of 
the material of which it is made ; 

Or, the value of E may be found by the elongation or bending of the 



We have here five quantities, which may be determined by experiment. 

on sphere. 



, audi: 


-i^ 


2 
3m 


We have here 


fiv 




(43) 


1. 




(42) 


2. 




(31) 


3. 




(17) 


4. 
5. 



+ — ) by external pressure 



Cj = ( j equal pressures. 

m by twisting the rod. 

/Xj the elasticity of the liquid. 



THE EQUILIBRIUM OF ELASTIC SOLIDS. 



55 



When the elastic sphere is solid, the internal radius a, vanishes, and 

fh=p = q, and -y = ^- 

When the case becomes that of a spherical cavity in an infinite solid, the 
external radius a^ becomes infinite, and 



P=K-f{K-K) 



r- 


= K+i 


^i'h-h,) 




r 


= ^^>i+^^(^>-^^) 


1 
m 


v = 


■'-! 







(44). 



The effect of pressure on the surface of a spherical cavity on any other part 
of an elastic solid is therefore inversely proportional to the cube of its distance 
from the centre of the cavity. 

When one of the surfaces of an elastic hollow sphere has its radius rendered 
invariable by the support of an incompressible sphere, whose radius is Oj, then 

— = 0, when r = a^, 



therefore 






2771 



q=h 



2a^m + 3«//x 

3a,V 
2a>i + 3a//x 



r* 2a^m + 3a//x 



IK 



W hen r = a,, j-y — lu r-—. ~— ,- 

" V -2a>2 + 3a.//i, 



K^ 



1 



r* 2a/m + 3a//i 



(45). 



Case V. 

On the equilibrium of an elastic beam of rectangular section uniformly 
bent. 

By supposing the bent beam to be produced till it returns into itself, we 
may treat it as a hollow cylinder. 



66 THE EQUILIBRIUM OF ELASTIC SOLIDS. 

Let a rectangular elastic beam, whose length is 2irc, be bent into a circular 
form, so as to be a section of a hollow cylinder, those parts of the beam which 
lie towards the centre of the circle will be longitudinally compressed, while the 
opposite parts will be extended. 

The expression for the tangential compression is therefore 

Br _ r — c 
r ~ c ' 

r 



Sr 
Comparing this value of — with that of Equation (20), 



V=(^-4)<''+-p+«)+^''' 



dr 

,,. , /I 2\ ., 

ion 



and by (21), q=p + r 

By substituting for q its value, and dividing by r (q- + ^) • the equat: 

becomes 

dp 2m + 3/x j9 _ 9?n/i. — {m — 3/x) o 9m/x c 
dr m + 6fx r~ (m + 6fi) r (m + 6/x) r' * 

a linear differential equation, which gives 

^ ^ m — 3fir 2m + 3/x 

Ci may be found by assumiQg that when r^a^, p = \, and q may be found 
from p by equation (21). 

As the expressions thus found are long and cumbrous, it is better to use 
the following approximations : — 

_/_9m^\ y ( ) 

l^\llcl^ \ (48). 

In these expressions a is half the depth of the beam, and y is the distance 
of any part of the beam from the neutral surface, which in this case is a cylin- 
dric surface, whose radius is c. 

These expressions suppose c to be large compared with a, since most sub- 
stances break when - exceeds a certain small quantity. 



THE EQUILIBRIUM OF ELASTIC SOLIDS. 57 

Let b be the breadth of the beam, then the force with which the beam 
resists flexure = M 



M=lhyq = ^^^-^ = Ef (49), 



which is the ordinary expression for the stiffness of a rectangular beam. 

The' stiffness of a beam of any section, the form of which is expressed by 
an equation between x and y, the axis of x being perpendicular to the plane of 
flexure, or the osculating plane of the axis of the beam at any point, is ex- 
pressed by 

Mc = E{ifdx (50), 

M being the moment of the force which bends the beam, and c the radius of 
the circle into which it is bent. 

Case YI. 

At the meeting of the British Association in 1839, Mr James Nasmyth 
described his method of making concave specula of silvered glass by bending. 

A circular piece of silvered plate-glass was cemented to the opening of an 
iron vessel, from which the air was afterwards exhausted. The mirror then 
became concave, and the focal distance depended on the pressure of the air. 

Buffon proposed to make burning- mirrors in this way, and to produce the 
partial vacuum by the combustion of the air in the vessel, which was to be 
effected by igniting sulphur in the interior of the vessel by means of a burn- 
ing-glass. Although sulphur evidently would not answer for this purpose, phos- 
phorus might; but the simplest way of removing the air is by means of the 
air-pump. The mirrors which were actually made by Buffon, were bent by 
means of a screw acting on the centre of the glass. 

To find an expression for the curvature produced in a flat, circular, elastic 
plate, by the difference of the hydrostatic pressures which act on each side 
of it,— 

Let t be the thickness of the plate, which must be small compared with 
its diameter. 

Let the form of the middle surface of the plate, after the curvature is 
produced, be expressed by an equation between r, the distance of any point 
from the axis, or normal to the centre of the plate, and x the distance of 
the point from the plane in which the middle of the plate originally was, and let 

ds=-^{dxY + {dr)\ 

VOL I. 8 



58 THE EQUILIBRIUM OF ELASTIC SOLIDS. 

Let A, be the pressure on one side of the plate, and h^ that on the other. 

Let p and q be the pressures in the plane of the plate at any point, p 
acting in the direction of a tangent to the section of the plate by a plane 
passing through the axis, and q acting in the direction perpendicular to that 
plane. 

By equating the forces which act on any particle in a direction parallel to 
the axis, we find 

^ drdx , ^ dpdx , ^ d^x ^ ,, j^dr 

By making p = when r = in this equation, when integrated, 

p-l^l^^--'^-) ("^- 

The forces perpendicular to the axis are 

[drV . dpdr , ^ d^r .^ i\dx ^ . 

Substituting for p its value, the equation gives 

_ (^1 - h^ idr dr dx\ (h^ - h^ /dr ds^d^^ds ^r\ , . 
^" t ''[d'sdi'^d^)'^ 2t "^^[didxd^ dxd^)""^ ^' 

The equations of elasticity become 

dSs (\ 1 \ / ^ h, + h\^p 

Differentiating -j- = -^ (""''')' ^^^ ^ ^^ *^^^® 

dhr dr dr dSs 

dr ~ ds ds ds ' 

By a comparison of these values of -t— , 



ds 

dr\ 
ds) \9iJ, 



, t^rwl 1\/ , ,K + h\,qdrp^ (I l\fdp,dq\ 



w dr as 



THE EQUILIBRIUM OF ELASTIC SOUDS. 59 

To obtain an expression for the curvature of the plate at the vertex, let a 

be the radius of curvature, then, as an approximation to the equation of the 

plate, let 

r» 
x — — . 
2a 

By substituting the value of a: in the values of p and q, and in the equa- 
tion of elasticity, the approximate value of a is found to be 






a = 



18m/x, \-\-h^ m- 3/x 

. 1 c 1 "T" ' T 7~ ~T~z ; — TT" 



.(53). 



^i-A, lOm + 51/x A,-^2 lOw + 51/t ' 

Since the focal distance of the mirror, or -, depends on the difference of 

pressures, a telescope on Mr Nasmyth's principle would act as an aneroid baro- 
meter, the focal distance varying inversely as the pressure of the atmosphere. 

Case VIL 

To find the conditions of torsion of a cylinder composed of a great number 
of parallel wires bound together without adhering to one another. 

Let X be the length of the cylinder, a its radius, r the radius at any point, 
hS the angle of torsion, M the force producing torsion, hx the change of length, 
and P the longitudinal force. Each of the wires becomes a helix whose radius 
is r, its angular rotation Zd, and its length along the axis x-Zx. 



Its length is therefore {rZey 



— IJ 



and the tension is = jE; 1 1 - /[ 1 - - ] V r^ (-]'] . 

This force, resolved parallel to the axis, is 



60 THE EQUIUBRTCM OF ELASTIC SOUDS. 

and since — and r — are small, we may assume 

XX 

-"-{-l-n?)'} <">■ 

The force, when resolved in the tangential direction, is approximately 

"-■^m'i-m '"> 

By eliminating — between (54) and (55) we have 

X 



M: ^^' 



^ip.E.^m (56). 

X 24 \ a?/ 



When P = 0, M depends on the sixth power of the radius and the cube 
of the angle of torsion, when the cylinder is composed of separate filaments. 

Since the force of torsion for a homogeneous cylinder depends on the 
fourth power of the radius and the first power of the angle of torsion, the 
torsion of a wire having a fibrous texture will depend on both these laws. 

The parts of the force of torsion which depend on these two laws may be 
found by experiment, and thus the difference of the elasticities in the direction 
of the axis and in the perpendicular directions may be determined. 

A calculation of the force of torsion, on this supposition, may be found in 
Young's Mathematical Principles of Natural Philosophy; and it \s introduced 
here to account for the variations from the law of Case II., which may be 
observed in a twisted rod. 



Case VIII. 

It is well known that grindstones and fly-wheels are often broken by the 
centrifugal force produced by their rapid rotation. I have therefore calculated 
the strains and pressure acting on an elastic cylinder revolving round its axis, 
and acted on by the centrifugal force alone. 



THE EQUILIBBIUM OF ELASTIC SOLIDa. 



61 



The equation of the equilibrium of a particle [see Equation (21)], becomes 



dp Air'k , 



where q and p are the tangential and radial pressures, k is the weight in 
pounds of a cubic inch of the substance, g is twice the height in inches that 
a body falls in a second, t is the time of revolution of the cylinder in seconds. 

By substituting the value of q and ^ in Equations (19), (20), and neglect- 
ing 0, 



-(i-3^)(«|-?-g)-M^S-f-^.^) 



which gives 



1 TT^k 






2gt^\ 



1 , Tj'k 



2+^K + ^« 



(-"?) 



TT'k 

2gf 



^=-V + 2^»(-2 + f)^ + c. 



(57). 



If the radii of the surfaces of the hollow cylinder be a, and cu„ and the 
pressures actmg on them h^ and h^, then the values of c^ and c, are 



(58). 



-f^'-(«--.')S(^-S. 

When o, = 0, as in the case of a solid cylinder, c, = 0, and 

« = *'+0 {2('^ + «.') + |(3'^-«,')} (59). 

When A, = 0, and r^a^, 

^ = ^U-2) (60). 

When q exceeds the tenacity of the substance in pounds per square inch, 
the cylinder will give way; and by making q equal to the number of pounds 
which a square inch of the substance will support, the velocity may be found 
at which the bursting of the cylinder will take place. 



g2 THE EQUILIBRIUM OP ELASTIC SOLIDS. 

Since I=ho>(q-p) = '^ (^-2\br', a transparent revolving cylinder, when 

polarized light is transmitted parallel to the axis, will exhibit rings whose 
diameters are as the square roots of an arithmetical progression, and brushes 
parallel and perpendicular to the plane of polarization. 

Case IX. 

A hollow cylinder or tube is surrounded by a medium of a constant 
temperature while a liquid of a different temperature is made to flow through 
it. The exterior and interior surfaces are thus kept each at a constant tem- 
perature till the transference of heat through the cylinder becomes uniform. 

Let V be the temperature at any point, then when this quantity has 



reached its limit, 



rdv _ 



v = Ci\ogr + Ci (61). 

Let the temperatures at the surfaces be 0^ and 0^, and the radii of the 
surfaces a, and a^, then 

^ 0^-0^ loga,0^-logaA 

^'""logaj-loga/ '~ loga^-loga^ 
Let the coeflBcient of linear dilatation of the substance be c,, then the 
proportional dilatation at any point will be expressed by c,v, and the equations 
of elasticity (18), (19), (20), become 

r \,9/x 3m/ ^ ^ ^' m 
The equation of equHibrivuu is 

2-P+r'^ (21), 

and since the tube is supposed to be of a considerable length 

-J— =c^ a constant quantity. 

CL2C 



THE EQUILIBRIUM OF ELASTIC SOLIDS. 63 

From these equations we find ttat 

9/x 3m 
and hence v = c^\ogr + Cz, p may be found in terms of r. 

Hence ? = (|l + 4) " ^.«' •«§ '- ^. ^ + <'• + (|l + ^) ''.^- 

Since I—hco (q —p) = ho)i— + - — ) CjCg — 260)05 -^ , 

the rings seen in this case will differ from those described in Case III. only 
by the addition of a constant quantity. 

When no pressures act on the exterior and interior surfaces of the tube 
^j = ^„ = 0, and 

/2 . J_V^.^ Aoo-r I ^i'^/ log^i-log«2 , a/logct,-a/logaA 



/^ 1_\- I a^a^ log g, - log ct , a^ log a, - a/ log a \ 

^-1,9,. + 3m/ ^^^3^^^S^ r^ a'-a^ + <-a,^ +V' 

\9/x 3m/ ' ' \ r" a{-a^ J 



...(62). 



There will, therefore, be no action on polarized light for the ring whose 
radius is r when 

r" = 2 „ log - . 



Case X. 

Sir David Brewster has observed {Edinburgh Transacticms, Vol. viii.), that 
when a solid cylinder of glass is suddenly heated at the cylindric siuface a 
polarizing force is developed, which is at any point proportional to the square 
of the distance from the axis of the cylinder ; that is to say, that the dif- 



64 THE EQUILIBBIUM OF ELASTIC SOLIDS. 

ference of retardation of the oppositely polari^ied rays of %ht is proportional 
to the square of the radius r, or 

/= bCj^cor' = h(o {q —p) = hayr -^ , 
Since if a be the radius of the cylinder, ^ = when r^a, 

Hence ?=J(3r'-o"). 

2 

By substituting these values of p and q in equations (19) and (20), and 
, . d h' dhr T ^ , 



^=|(4 + li)'-' + »" (««)• 



c^ being the temperature of the axis of the cylinder, and c, the coefficient of 
linear expansion for glass. 

Case XI. 

Heat is passing uniformly through the sides of a spherical vessel, such as 
the ball of a thermometer, it is required to determine the mechanical state of 
the sphere. As the methods are nearly the same as in Case IX., it will be 
sufficient to give the results, using the same notation. 

, dv c, 

dr ^' * r 

Ci = aM,— ?, c- = -5-2 —, 

o, — o, o, — a, 

1 /2 .1 \-^ 1 . 

When h, = h, = the expression for p becomes 

p = /2 ly- r_aXLl _^A.l^ a.'-a» | 

^ \9/t* 3m/ '^ ' ''[a/-a/7^ a,-o^r {0,-0,) (o^-o^)] ^ ' 

From this value of p the other quantities may be found, as in Case IX., 
from the equations of Case IV. 



THE EQUILIBRIUM OF ELASTIC SOLIDS. 65 



Case XII. 

When a long beam is bent into the form of a closed circular ring (as in 
Case v.), all the pressures act either parallel or perpendicular to the direction 
of the length of the beam, so that if the beam were divided into planks, there 
would be no tendency of the planks to slide on one another. 

But when the beam does not form a closed circle, the planks into which it 
may be supposed to be divided will have a tendency to slide on one another, 
and the amount of sliding is determined by the linear elasticity of the sub- 
stance. The deflection of the beam thus arises partly from the bending of the 
whole beam, and partly from the sHding of the planks ; and since each of these 
deflections is small compared with the length of the beam, the total deflection 
will be the sum of the deflections due to bending and sliding. 



Let 



A=Mc = E\xi/'dy (65). 



A is the stiffiiess of the beam as found in Case Y., the equation of the 
transverse section being expressed in terms of x and y, y being measured from 
the neutral surface. 

Let a horizontal beam, whose length is 2l, and whose weight is 2w, be 
supported at the extremities and loaded at the middle with a weight W. 

Let the deflection at any point be expressed by h^, and let this quantity 
be small compared with the length of the beam. 

At the middle of the beam, 8,y is found by the usual methods to be 

% = ^ {-h^w + ^^l'W) (66). 



Let 



B = — \xdy = — (sectional area) (jo7). 



B is the resistance of the beam to the sliding of the planks. The de- 
flection of the beam arising from this cause is 

% = 2]b(^'+^^ (68). 

VOL. I. 9 



66 THE EQXnUBRnJM OF ELASTIC SOLIDS. 

This quantity is small compared with S^y, when the depth of the beam is 
small compared with its length. 

The whole deflection ^y = B^ + S^ 



A3/ = - (^.Z-^iS + ^ {U +^l) (^^)- 



Case XIII. 

When the values of the compressions at any point have been found, when 
two difierent sets of forces act on a solid separately, the compressions, when 
the forces act at the same time, may be found by the composition of com- 
pressions, because the small compressions are independent of one another. 

It appears from Case I., that if a cylinder be twisted as there described, 
the compressions will be inversely proportional to the square of the distance 
from the centre. 

If two cylindric surfaces, whose axes are perpendicular to the plane of an 
indefinite elastic plate, be equally twisted in the same direction, the resultant 
compression in any direction may be found by adding the compression due to 
each resolved in that direction. 

The result of this operation may be thus stated geometrically. Let A^ and 
A^ (Fig. 1) be the centres of the twisted cylinders. Join ^1^25 and bisect A^A, 
in 0. Draw OBC at right angles, and cut off OB^^ and OB^ each equal to OA^. 

Then the difference of the retardation of oppositely polarized rays of light 
passing perpendicularly through any point of the plane varies directly as the 
product of its distances from B^ and B^, and inversely as the square of the 
product of its distances from A^ and A^. 

The isochromatic lines are represented in the figure. 

The retardation is infinite at the points ^1 and A^; it vanishes at B^^ 
and jBj ; and if the retardation at be taken for unity, the isochromatic curves 
2, 4, surround Aj^ and A^; that in which the retardation is unity has two 
loops, and passes through 0; the curves ^, ^ are continuous, and have points 
of contrary flexure ; the curve ^ has multiple points at Cj and C,, where 



THE EQUILIBEIUM OF ELASTIC SOLIDS. 



67 



.4,(7, = -4,^,, and two loops surrounding B^ and B^', the other curves, for which 
/=l4-» -gS-j ^c-» consist each of two ovals surrounding B^ and jB,, and an 
exterior portion surrounding all the former curves. 

Fig. 1. 




I have produced these curves in the jelly of isinglass described in Case I. 
They are best seen by using circularly polarised light, as the curves are then 
seen without interruption, and their resemblance to the calculated curves is 
more apparent. To avoid crowding the curves toward the centre of the figure, 
I have taken the values of / for the different curves, not in an arithmetical, 
but in a geometrical progression, ascending by powers of 2. 



68 



THE EQUILrBRTOM OF ELASTIC SOLIDS. 



Case XIV. 

On the determination of the pressures which act in the interior of trans- 
parent solids, from observations of the action of the solid on polarized light. 

Sir David Brewster has pointed out the method by which polarized light 
might be made to indicate the strains in elastic solids ; and his experiments on 
bent glass confirm the theories of the bending of beams. 

The phenomena of heated and unannealed glass are of a much more complex 
nature, and they cannot be predicted and explained without a knowledge of the 
laws of cooling and solidification, combined with those of elastic equilibrium. 

In Case X. I have given an example of the inverse problem, in the case 
of a cylinder in which the action on light followed a simple law ; and I now 
go on to describe the method of determuiing the pressures in a general case, 
applying it to the case of a triangle of unannealed plate-glass. 



D D 




Fig. 3. 



The lines of equal intensity of the action on Hght are seen without 
interruption, by using circularly polarized light. They are represented in Fig. 2, 
where A, BBB, DDD are the neutral points, or points of no action on light, 
and CCC, EEE are the points where that action is greatest ; and the intensity 



THE EQUILIBRIUM OF ELASTIC SOLIDS. 69 

of the action at any other point is determined by its position with respect to 
the isochromatic curves. 

The direction of the principal axes of pressure at any point is found by 
transmitting plane polarized light, and analysing it in the plane perpendicular 
to that of polarization. The light is then restored in every part of the triangle, 
except in those points at which one of the principal axes is parallel to the 
plane of polarization. A dark band formed of all these points is seen, which 
shifts its position as the triangle is turned round in its own plane. Fig. 3 
represents these curves for every fifteenth degree of inclination. They correspond 
to the lines of equal variation of the needle in a magnetic chart. 

From these curves others may be found which shall indicate, by their own 
direction, the direction of the principal axes at any point. These curves of 
direction of compression and dilatation are represented in Fig. 4 ; the curves 
whose direction corresponds to that of compression are concave toward the 
centre of the triangle, and intersect at right angles the curves of dilatation. 

Let the isochromatic lines in Fig. 2 be determined by the equation 

<^,{x,y) = I- = (o{q-p)-, 

where / is the difference of retardation of the oppositely polarized rays, and 
q and p the pressures in the principal axes at any point, z being the thick- 
ness of the plate. 

Let the lines of equal inclination be determined by the equation 

<^2 (^. y) = tan 6, 

6 being the angle of inclination of the principal axes ; then the differential 
equation of the curves of direction of compression and dilatation (Fig. 4) is 

By considering any particle of the plate as a portion of a cylinder whose 
axis passes through the centre of curvature of the curve of compression, we find 

?-?>=^^ (21). 



70 THE EQUILIBRIUM OF EliASTIC SOLIDS. 

Let R denote the radius of curvature of the curve of compression at any 
point, and let S denote the length of the curve of dilatation at the same 
point, 

and since {q -p), R and S are known, and since at the surface, where (^^ {x, y) = 0, 
j9 = 0, all the data are given for determining the absolute value of p by inte- 
gration. 

Though this is the best method of finding p and q by graphic construc- 
tion, it is much better, when the equations of the curves have been found, that 
is, when ^i and <j>^ are known, to resolve the pressures in the direction of the 
axes. 

The new quantities are p^, p„ and ^3 ; and the equations are 

tan^=-^, {p-qY = q.' + (p.-p.y, Pi+P.=P + q- 
Pi Pi 

It is therefore possible to find the pressures from the curves of equal tint 
and equal inclination, in any case in which it may be required. In the mean- 
time the curves of Figs. 2, 3, 4 shew the correctness of Sir John Herschell's 
ingenious explanation of the phenomena of heated and unannealed glass. 



Note A. 

As the mathematical laws of compressions and pressures have been very thoroughly 
investigated, and as they are demonstrated with great elegance in the very complete and 
elaborate memoir of MM. Lamd and Clapeyron, I shall state as briefly as possible their results. 

Let a solid be subjected to compressions or pressures of any kind, then, if through any 
point in the solid lines be drawn whose lengths, measured from the given point, are pro- 
portional to the compression or pressure at the point resolved in the directions in which the 
lines are drawn, the extremities of such lines will be in the surface of an ellipsoid, whose 
centre is the given point. 

The properties of the system of compressions or pressures may be deduced from those 
of the ellipsoid. 



THE EQUILIBRIUM OF ELASTIC SOLIDS. 71 

There are three diameters having perpendicular ordinates, which are called the principal 
axes of the ellipsoid. 

Similarly, there are always three directions in the compressed particle in which there 
is no tangential action, or tendency of the parts to slide on one another. These directions 
are called the principal axes of compression or of pressure, and in homogeneous solids they 
always coincide with each other. 

The compression or pressure in any other direction is equal to the sum of the products 
of the compressions or pressures in the principal axes multiplied into the squares of the 
cosines of the angles which they respectively make with that direction. 



Note B. 

The fundamental equations of this paper differ from those of Navier, Poisson, &c., only 
in not assuming an invariable ratio between the linear and the cubical elasticity; but since 
I have not attempted to deduce them from the laws of molecular action, some other reasons 
must be given for adopting them. 

The experiments from which the laws are deduced are — 

1st. Elastic solids put into motion vibrate isochronously, so that the sound does not 
vary with the amplitude of the vibrations. 

2nd. Regnault's experiments on hollow spheres shew that both linear and cubic com- 
pressions are proportional to the pressures. 

3rd. Experiments on the elongation of rods and tubes immersed in water, prove that 
the elongation, the decrease of diameter, and the increase of volume, are proportional to the 
tension. 

4th. In Coulomb's balance of torsion, the angles of torsion are proportional to the 
twisting forces. 

It would appear from these experiments, that compressions are always proportional to 
pressures. 

Professor Stokes has expressed this by making one of his coefficients depend on the 
cubical elasticity, Avhile the other is deduced from the displacement of shifting produced by 
a given tangential force. 

M. Cauchy makes one coefficient depend on the linear compression produced by a force 
acting in one direction, and the other on the change of volume produced by the same force. 

Both of these methods lead to a correct result ; but the coefficients of Stokes seem to 
have more of a real signification than those of Cauchy ; I have therefore adopted tiiose of 
Stokes, using the symbols m and fi, and the fundamental equations (4) and (5), which define 
them. 



72 



THE EQUILIBRIUM OF ELASTIC SOLIDS. 



Note C. 

As the coefficient <w, which determines the optical effect of pressure on a substance, 
varies from one substance to another, and is probably a function of the linear elasticity, a 
determination of its value in different substances might lead to some explanation of the 
action of media on light. 



This paper commenced by pointing out the insufficiency of all theories of elastic solids, 
in which the equations do not contain two independent constants deduced from experiments. 
One of these constants is common to liquids and solids, and is called the modulus of cubical 
elasticity. The other is peculiar to solids, and is here called the modulus of linear elasticity. 
The equations of Navier, Poisson, and Lam^ and Clapeyron, contain only one coefficient; 
and Professor G. G. Stokes of Cambridge, seems to have formed the first theory of elastic 
solids which recognised the independence of cubical and linear elasticity, although M. Cauchy 
seems to have suggested a modification of the old theories, which made the ratio of linear 
to cubical elasticity the same for all substances. Professor Stokes has deduced the theory 
of elastic solids from that of the motion of fluids, and his equations are identical with those 
of this paper, which are deduced from the two following assumptions. 

In an element of an elastic solid, acted on by three pressures at right angles to one 
another, as long as the compressions do not pass the limits of perfect elasticity — 

1st. The sum of the pressures, in three rectangular axes, is proportional to the sum 
of the compressions in those axes. 

2nd. The difference of the pressures in two axes at right angles to one another, is 
proportional to the difference of the compressions in those axes. 



Or, in symbols: 



(P. + P..i'J = 3.(^%|4). 



(^.-^.)=-(l 



(P, 



p,)=r,j'y. 



(P,-P^ = m 



fZz Bx 



fi being the modulus of auhical, and m that of linear elasticity. 

These equations are found to be very convenient for the solution of problems, some 
of which were given in the latter part of the paper. 



THE EQUILIBRIUM OF ELASTIC SOLIDS. 73 

These particular cases were — 

That of an elastic hollow cylinder, the exterior surface of which was fixed, while the 
interior was turned through a small angle. The action of a transparent solid thus twisted 
on polarized light, was calculated, and the calculation confirmed by experiment. 

The second case related to the torsion of cylindric rods, and a method was given by 

which m may be found. The quantity E= ^ was found by elongating, or by bending 

the rod used to determine m, and fi is found by the equation, 

_ Em 
^~dm-6E' 

The effect of pressure on the surfaces of a hollow sphere or cylinder was calculated, 
and the result applied to the determination of the cubical compressibility of liquids and 
solids. 

An expression was found for the curvature of an elastic plate exposed to pressure on 
one side ; and the state of cylinders acted on by centrifugal force and by heat was 
determined. 

The principle of the superposition of compressions and pressures was applied to the case of 
a bent beam, and a formula was given to determine E from the deflection of a beam 
supported at both ends and loaded at the middle. 

The paper concluded with a conjecture, that as the quantity a (which expresses the 
relation of the inequality of pressure in a solid to the doubly-refracting force produced) is 
probably a function of m, the determination of these quantities for different substances 
might lead to a more complete theory of double refraction, and extend our knowledge of the 
laws of optics. 



VOL. I. 10 



[Extracted from the Cambridge and Dublin Mathematical Journal, Vol. viii. p. 188, 

February/, 1854.] 



Solutions of Problems. 



1. If from a point in the circumference of a vertical circle two heavy particles be suc- 
cessively projected along the curve, their initial velocities being equal and either in the same 
or in opposite directions, the subsequent motion will be such that a straight line joining 
the particles at any instant will touch a circle. 

Note. The particles are supposed not to interfere with each other's motion. 



The direct analytical proof would involve the properties of elliptic integrals, 
but it may be made to depend upon the following geometrical theorems. 

(1) If from a point in one of two circles a right line be drawn cutting 
the other, the rectangle contained by the segments so formed is double of the 
rectangle contained by a line drawn from the point perpendicular to the radical 
axis of the two circles, and the line joining their centres. 

The radical axis is the line joining the points of intersection of the two 
circles. It is always a real hne, whether the points of intersection of the circles 
be real or imaginary, and it has the geometrical property — that if from any point 
on the radical axis, straight lines be drawn cutting the circles, the rectangle con- 
tained by the segments formed by one of the circles is equal to the rectangle 
contained by the segments formed by the other. 

The analytical proof of these propositions is very simple, and may be resorted 
to if a geometrical proof does not suggest itself as soon as the requisite figure 
is constructed. 

If ^, B be the centres of the circles, P the given point in the circle whose 
centre is ^, a line drawn from P cuts the first circle in p, the second in Q 



SOLUTIONS OF PROBLEMS. 75 

and q, and the radical axis in R. If PH be drawn perpendicular to the radical 
axis, then 

PQ.Pq = 2AB.HP. 

CoR. If the line be drawn from P to touch the circle in T, instead of 
cutting it in Q and q, then the square of the tangent PT is equal to the 
rectangle 2AB . HP. 

Similarly, if ph be drawn from p perpendicular to the radical axis 

p'P = 2AB.hp. 

Hence, if a line be drawn touching one circle in T, and cutting the other 
in P and p, then 

(PTY : {pT)' :: HP : hp. 

(2) If two straight lines touching one circle and cutting another be made 
to approach each other indefinitely, the small arcs intercepted by their inter- 
sections with the second circle wiU be ultimately proportional to their distances 
from the point of contact. 

This result may easily be deduced from the properties of the similar 
triangles FTP and ppT. 

Cor. If particles P, p be constrained to move in the circle A, while 
the line Pp joining them continually touches the circle B, then the velocity 
of P at any instant is to that of p as PT to pT ; and conversely, if the 
velocity of P at any instant be to that of P as PT to pT, then the line 
Pp will continue to be a tangent to the circle B. 

Now let the plane of the circles be vertical and the radical axis horizontal, 
and let gravity act on the particles P, p. The particles were projected from 
the same point with the same velocity. Let this velocity be that due to the 
depth of the point of projection below the radical axis. Then the square of 
the velocity at any other point will be proportional to the perpendicular from 
that point on the radical axis ; or, by the corollary to (l), if P and p be at 
any time at the extremities of the line PTp, the square of the velocity of P 
will be to the square of the velocity of p as PH to ph, that is, as (PTf to 
(pTf. Hence, the velocities of P and p are in the proportion of PT to pT, 
and therefore, by the corollary to (2), the line joining them will continue a 
tangent to the circle B during each instant, and will therefore remain a tangent 
during the motion. 



76 SOLUTIONS OF PROBLEMS. 

The cb'cle A, the radical axis, and one position of the line Pp, are given 
by the circumstances of projection of P and p. From these data it is easy to 
determine the circle jB by a geometrical construction. 

It is evident that the character of the motion will determine the position 
of the circle B. If the motion is oscillatory, B will intersect A. If P and p 
make complete revolutions in the same direction, B will lie entirely within A, 
but if they move in opposite directions, B will lie entirely above the radical axis. 

If any number of such particles be projected from the same point at equal 
intervals of time with the same direction and velocity, the lines joining successive 
particles at any instant will be tangents to the same circle ; and if the time 
of a complete revolution, or oscillation, contain n of these intervals, then these 
lines will form a polygon of ?i sides, and as this is true at any instant, any 
number of such polygons may be formed. 

Hence, the following geometrical theorem is true : 

"If two circles be such that n lines can be drawn touching one of them 
and having their successive intersections, including that of the last and first, 
on the circiunference of the other, the construction of such a system of lines 
wiU be possible, at whatever point of the first circle we draw the first tangent." 



2. A transparent medium is such that the path of a ray of light within it is a given 
circle, the index of refraction being a function of the distance from a given point in the 
plane of the circle. 

Find the form of this function and shew that for light of the same refrangibility — 

(1) The path of every ray witJdn the medium is a circle, 

(2) All the rays proceeding from any point in the medium will meet accurately in 
another point. 

(3) If rays diverge from a point without the medium and enter it through a spherical 
surface having that point for its centre, they will be made to converge accurately to a point 
within the medium. 

Lemma I. Let a transparent medium be so constituted, that the refractive 
index is the same at the same distance from a fixed point, then the path of 
any ray of light within the medium will be in one plane, and the perpen- 



SOLUTIONS OF PROBLEMS. 77 

dicular from the fixed point on the tangent to the path of the ray at any 
point will vary inversely as the refractive index of the medium at that point. 

We may easily prove that when a ray of light passes through a spherical 
surface, separating a medium whose refractive index is /x, from another where 
it is /Aj, the plane of incidence and refraction passes through the centre of 
the sphere, and the perpendiculars on the direction of the ray before and after 
refraction are ir the ratio of /i, to fi^. Since this is true of any number of 
spherical shells of different refractive powers, it is also true when the index of 
refraction varies continuously from one shell to another, and therefore the 
proposition is true. 

Lemma II. If from any fixed point in the plane of a circle, a perpen- 
dicular be drawn to the tangent at any point of the circumference, the rectangle 
contained by this perpendicular and the diameter of the circle is equal to the 
square of the line joining the point of contact with the fixed point, together 
with the rectangle contained by the segments of any chord through the fixed 
point. 

Let APB be the circle, the fixed point; then 
OY.FE=OP' + AO.OB, 




Produce PO to Q, and join QR, then the triangles OYP, PQR are similar; 
therefore 

OY.PR=OP.PQ 

= OP' + OP.OQ; 
.: OY.PR=OP' + AO.OB. 
If we put in this expression AO . OB = a^, 

PO = r, OY=p, PR = 2p, 
it becomes 2pp = ?'*+■ a*, 



78 SOLUTIONS OF PROBLEMS. 

To find the law of the index of refraction of the medium, so that a ray 
from A may describe the circle APB, /x must be made to vary inversely as p 
by Lemma I. 

Let AO = r^, and let the refractive index at A=fii; then generally 



h' 


_c 
p 


_ 2C7p . 
a' + r'' 




/^1 = 


. 2Cp 

a' + r:' 




a' + r,' 



but at A 

therefore 

The value of /n at any point is therefore independent of p, the radius of 

the given circle; so that the same law of refractive index will cause any other 

ray to describe another circle, for which the value of a' is the same. The 

a^ . . . 

value of OB is — , which is also independent of p ; so that every ray which 

proceeds from A must pass through B. 

Again, if we assume /x^ as the value of /x when r = 0, 

ar + r,' 



therefore h' — H-o 



d' + r'^-' 



a result independent of r^. This shews that any point A' may be taken as 
the origin of the ray instead of A, and that the path of the ray will still be 
circular, and will pass through another point B' on the other side of 0, such that 

Next, let CP be a ray from C, a point without the medium, falling at P 
on a spherical surface whose centre is C. 

Let be the fixed point in the medium as before. Join PO, and produce 

to Q till OQ = jyp. Through Q draw a circle touching CP in P, and cutting 

CO in A and B ; then PBQ is the path of the ray within the medium. 



SOLUTIONS OF PROBLEMS. 79 




Since CP touches the circle, we have 
CP'^CA. CB, 

= {CO-OA){CO-\-OB); 

but 0A= -^; 

therefore CF' = CQ + CO (oB - ^^ 

an equation whence OB may be found, B being the point in the medium 
through which all rays from C pass. 

Note. The possibility of the existence of a medium of this kind possessing 
remarkable optical properties, was suggested by the contemplation of the structure 
of the crystalline lens in fish; and the method of searching for these properties 
was deduced by analogy from Newton's Principia, Lib. L Prop. vii. 

It would require a more accurate investigation into the law of the refractive 
index of the different coats of the lens to test its agreement with the supposed 
medium, which is an optical instrument theoretically perfect for homogeneous 
light, and might be made achromatic by proper adaptation of the dispersive 
power of each coat. 

On the other hand, we find that the law of the index of refraction which 
would give a minimum of aberration for a sphere of this kind placed in water, 
gives results not discordant with facts, so far as they can be readily ascertained. 



[From the Transactions of the Cambridge Philosophical Society, Vol. ix. Part iv.] 



IV. On the Transformation of Surfaces by Bending. 



Euclid has given two definitions of a surface, which may be taken as 
examples of the two methods of investigating their properties. 

That in the first book of the Elements is — 

"A superficies is that which has only length and breadth." 

The superficies difiers from a line in having breadth as well as length, 
and the conception of a third dimension is excluded without being expHcitly 
introduced. 

In the eleventh book, where the definition of a soHd is first formally 
given, the definition of the superficies is made to depend on that of the solid — 
" That which bounds a soHd is a superficies." 

Here the conception of three dimensions in space is employed in forming 
a definition more perfect than that belonging to plane Geometry. 

In our analytical treatises on geometry a surface is defined by a function 
of three independent variables equated to zero. The surface is therefore the 
boundary between the portion of space in which the value of the function is 
positive, and that in which it is negative; so that we may now define a 
surface to be the boundary of any assigned portion of space. 

Surfaces are thus considered rather with reference to the figures which they 
limit than as having any properties belonging to themselves. 

But the conception of a surface which we most readily form is that of 
a portion of matter, extended in length and breadth, but of which the thick- 



TKANSFORMATION OF SURFACES BY BENDING. 81 

ness may be neglected. By excluding the thickness altogether, we arrive at 
Euclid's first definition, which we may state thus — 

" A surface is a lamina of which the thickness is diminished so as to become 
evanescent." 

We are thus enabled to consider a surface by itself, without reference to 
the portion of space of which it is a boundary. By drawing figures on the 
surface, and investigating their properties, we might construct a system of 
theorems, which would be true independently of the position of the surface in 
space, and which might remain the same even when the form of the solid of 
which it is the boundary is changed. 

When the properties of a surface with respect to space are changed, while 
the relations of lines and figures in the surface itself are unaltered, the surface 
may be said to preserve its identity, so that we may consider it, after the 
change has taken place, as the same surface. 

When a thin material lamina is made to assume a new form it is said 
to be hent. In certain cases this process of bending is called development, and 
when one surface is bent so as to coincide with another it is said to be 
applied to it. 

By considering the lamina as deprived of rigidity, elasticity, and other 
mechanical properties, and neglecting the thickness, we arrive at a mathemati- 
cal definition of this kind of transformation. 

" The operation of bending is a continuous change of the form of a surface, 
without extension or contraction of any part of it." 

The following investigations were undertaken with the hope of obtaining 
more definite conceptions of the nature of such transformations by the aid of 
those geometrical methods which appear most suitable to each particular case. 
The order of arrangement is that in which the different parts of the subject 
presented themselves at first for examination, and the methods employed form 
parts of the original plan, but much assistance in other matters has been 
derived from the works of Gauss*, Liouvillef, Bertrand^, Puiseux§, &c., references 
to which will be given in the course of the investigation. 

* Disquisitiones generalea circa superficies curvas. Presented to the Royal Society of Gottingen, 
8th October, 1827. Commentationes Recentiores, Tom. vi. 

t Liouville's Journal, xii. X ^^'^- ^^'^' § ^^"^■ 

VOL, I. 11 



82 



TRANSFORMATION OF SURFACES BY BENDING. 



On the Bending of Surfaces generated hy the motion of a straight line in space. 



If a straight line can be drawn in any surface, we may suppose that 
part of the surface which is on one side of the straight line to be fixed, 
while the other part is turned about the straight line as an axis. 

In this way the surface may be bent about any number of generating lines 
as axes successively, till the form of every part of the surface is altered. 

The mathematical conditions of this kind of bending may be obtained in 
the following manner. 

Let the equations of the generating line be expressed so that the constants 
involved in them are functions of one independent variable u, by the variation of 
which we pass from one position of the line to another. 

If in the equations of the generating line Aa, u = u^, then in the equations 
of the line Bh we may put u = U2, and from the equations of these lines we 
may find by the common methods the equations of the shortest line PQ between 
Aa and Bb, and its length, which we may call S^. We may also find the 
angle between the directions of ^a and Bb, and let this angle be SO. 

In the same way from the equations of 
Cc, in which u = u^, we may deduce the equa- 
tions of RS, the shortest line between Bb and 
Cc, its length 8^5 and the angle hd^ between 
the directions of Bb and Cc. We may also 
find the value of QR, the distance between 
the points at which PQ and RS cut Bb. 
Let QR = h(T, and let the angle between the 
directions of PQ and RS be S^. 

Now suppose the part of tlie surface between the lines Aa and Bb to be 
fixed, while the part between Bb and Cc is turned round Bb as an axis. The 
line RS wiU then revolve round the point R, remaining perpendicular to Bhy 
and Cc will still be at the same distance from Bb, and wiU make the same 
angle with it. Hence of the four quantities S4j S^2> ^cr and 8</>, 8^ alone will 
be changed by the process of bending. 8<^, however, may be varied in a 
perfectly arbitrary manner, and may even be made to vanish. 



,•?_.. 




TRANSFORMATION OF SURFACES BY BENDING. 83 

For, PQ and RS being both perpendicular to Bh, RS may be turned 
about Bh till it is parallel to PQ, in which case 8^ becomes = 0. 

By repeating this process, we may make all the " shortest lines" parallel to 
one another, and then all the generating lines will be parallel to the same 
plane. 

We have hitherto considered generating lines situated at finite distances from 
one another ; but what we have proved will be equally true when their distances 
are indefinitely diminished. Then in the limit 





du 


B0 


dO 


u,-u, 


" du 


Str 


da- 

" du 


8(f> 


d(f> 



Uj — Wi '* du ' 

All these quantities being functions of u, ^, 0, a- and (f), are functions of u 
and of each other; and if the forms of these functions be known, the positions 
of all the generating lines may be successively determined, and the equation 
to the surface may be found by integrating the equations containing the values 
of ^, 0, a- and <j). 

When the surface is bent in any manner about the generating lines, C> ^, 
and a- remain unaltered, but cf) is changed at every point. 

The form of <^ as a function of u will depend on the nature of the 
bending ; but since this is perfectly arbitrary, <^ may be any arbitrary function 
of u. In this way we may find the form of any surface produced by bending 
the given surface along its generating lines. 

By making <f) = 0, we make all the generating lines parallel to the same 
plane. Let this plane be that of xy, and let the first generating line coincide 
with the axis of x, then C will be the height of any other generating line 
above the plane of xy, and the angle which its projection on that plane 
makes with the axis of x. The ultimate intersections of the projections of the 
generating lines on the plane of xy will form a curve, whose length, measured 
from the axis of x, will be o-. 



84 TRANSFORMATION OF SURFACES BY BENDING. 

Since ia this case the quantities C> ^, and cr are represented bj distinct 
geometrical quantities, we may simplify the consideration of all surfaces generated 
by straight lines by reducing them by bending to the case in which those lines 
are parallel to a given plane. 

In the class of surfaces in which the generating lines ultimately intersect, 

-T- = 0, and ^ constant. If these surfaces be bent so that <j> = 0, the whole of 

the generating lines will lie in one plane, and their ultimate intersections will 
form a plane curve. The surface is thus reduced to one plane, and therefore 
belongs to the class usually described as "developable surfaces." The form of a 
developable surface may be defined by means of the three quantities 0, a- and 
(f>. The generating lines form by their ultimate intersections a curve of double 
curvature to which they are all tangents. This curve has been called the 
cuspidal edge. The length of this curve is represented by a, its absolute 

curvature at any point by -j- , and its torsion at the same point by ■— . 

When the surface is developed, the cuspidal edge becomes a plane curve, 
and every part of the surface coincides with the plane. But it does not follow 
that every part of the plane is capable of being bent into the original form 
of the surface. This may be easily seen by considering the surface when the 
position of the cuspidal edge nearly coincides with the plane curve but is not 
confounded with it. It is evident that if from any point in space a tangent 
can be drawn to the cuspidal edge, a sheet of the surface passes through that 
point. Hence the number of sheets which pass through one point is the same 
as the number of tangents to the cuspidal edge which pass through that 
point ; and since the same is true in the limit, the number of sheets which 
coincide at any point of the plane is the same as the number of tangents 
which can be drawn from that point to the plane curve. In constructing a 
developable surface of paper, we must remove those parts of the sheet from 
which no real tangents can be drawn, and provide additional sheets where more 
than one tangent can be drawn. 

In the case of developable surfaces we see the importance of attending to 
the position of the lines of bending; for though all developable surfaces may 
be produced from the same plane surface, their distinguishing properties depend 
on the form of the plane curve which determines the lines of bending. 



TRANSFORMATION OF SURFACES BY BENDING. 



85 



II. 

On the Bending of Surfaces of Revolution. 

In the cases previously considered, the bending in one part of the surface 
may take place independently of that in any other part. In the case now 
before us the bending must be simultaneous over the whole surface, and its 
nature must be investigated by a different method. 

The position of any point P on a surface of revolution may be deter- 
mined by the distance FV from the vertex, measured 
along a generating line, and the angle AVO which 
the plane of the generating line makes with a fixed 
plane through the axis. Let FV=s and AVO = 6. 
Let r be the distance {Pp) of P from the axis ; r 
will be a function of s depending on the form of the 
generating curve. 

Now consider the small rectangular element of the surface at P. Its length 
PR = Ss, and its breadth PQ = rhd, where r is a function of s. 

If in another surface of revolution r is some other function of s, then the 
length and breadth of the new element will be hs and rB$', and if 




r = /xr, and 0' = -0, 

rze'=rze, 

and the dimensions of the two elements will be the same. 

Hence the one element may be applied to the other, and the one surface 
may be applied to the other surface, element to element, by bending it. To 
effect this, the surface must be divided by cutting it along one of the generating 
lines, and the parts opened out, or made to overlap, according as /x is greater 
or less than unity. 

To find the effect of this transformation on the form of the surface we 
must find the equation to the original form of the generating line in terms of 
6" and r, then putting / = /ir, the equation between s and r will give the form 
of the generating line after bending. 



86 TRANSFORMATION OF SURFACES BY BENDING. 

When /x is greater than 1 it may happen that for some values of 5, y- is 

greater than -. In this case 

-j- = fi-j- is greater than 1 ; 

a result which indicates that the curve becomes impossible for such values of 
s and ft. 

The transformation is therefore impossible for the corresponding part of 
the surface. If, however, that portion of the original surface be removed, the 
remainder may be subjected to the required transformation. 



The theory of bending when apphed to the case of surfaces of revolution 
presents no geometrical difficulty, and little variety; but when we pass to 
the consideration of surfaces of a more general kind, we discover the insufficiency 
of the methods hitherto employed, by the vagueness of our ideas with respect 
to the nature of bending in such cases. In the former case the bending is 
of one kind only, and depends on the variation of one variable ; but the 
surfaces we have now to .consider may be bent in an infinite variety of ways, 
depending on the variation of three variables, of which we do not yet know the 
nature or interdependence. 

We have therefore to discover some method sufficiently general to be appli- 
cable to every possible case, and yet so definite as to limit each particular case 
to one kind of bending easily imderstood. 

The method adopted in the following investigations is deduced from the 
consideration of the surface as the limit of the inscribed polyhedron, when the 
size of the sides is indefinitely diminished, and their number indefinitely increased. 

A method is then described by which such a polyhedron may be inscribed 
in any surface so that all the sides shall be triangles, and aU the solid angles 
composed of six plane angles. 

The problem of the bending of such a polyhedron is a question of trigo- 
nometry, and equations might be found connecting the angles of the different 
edges which meet in each soHd angle of the polyhedron. It will be shewn that 



TRANSFORMATION OF SURFACES BY BENDING. 87 

the conditions thus obtained would be equivalent to three equations between 
the six angles of the edges belonging to each solid angle. Hence three addi- 
tional conditions would be necessary to determine the value of every such angle, 
and the problem would remain as indefinite as before. But if by any means 
we can reduce the number of edges meeting in a point to four, only one con- 
dition would be necessary to determine them all, and the problem would be 
reduced to the consideration of one kind of bending only. 

This may be done by drawing the polyhedron in such a manner that the 
planes of adjacent triangles coincide two and two, and form quadrilateral facets, 
four of which meet in every solid angle. The bending of such a polyhedron 
can take place only in one way, by the increase of the angles of two of the 
edges which meet in a point, and the diminution of the angles of the other two. 

The condition of such a, polyhedron being inscribed in any surface is then 
found, and it is shewn that when two forms of the same surface are given, 
a perfectly definite rule may be given by which two corresponding polyhedrons 
of this kind may be inscribed, one in each surface. 

Since the kind of bending completely defines the nature of the quadrilateral 
polyhedron which must be described, the lines formed by the edges of the 
quadrilateral may be taken as an indication of the kind of bending performed 
on the surface. 

These lines are therefore defined as " Lines of Bending." 

When the lines of bending are given, the forms of the quadrilateral facets 
are completely determined ; and if we know the angle which any two adjacent 
facets make with one another, we may determine the angles of the three edges 
which meet it at one of its extremities. From each of these we may find the 
angles of three other edges, and so on, so that the form of the polyhedron 
after bending will be completely determined when the angle of one edge is given. 
The bending is thus made to depend on the change of one variable only. 

In this way the angle of any edge may be calculated from that of any 
given edge ; but since this may be done in two different ways, by passing 
along two different sets of edges, we must have the condition that these results 
may be consistent with each other. This condition is satisfied by the method 
of inscribing the polyhedron. Another condition will be necessary that tlie 
change of the angle of any edge due to a small change of the given angle, 
produced by bending, may be the same by both calculations. This is the con- 
dition of " Instantaneous Lines of Bending." That tliis condition mav ccntinue 



88 TRANSFORMATION OF SURFACES BY BENDING. 

to be satisfied during the whole process we must have another, which is the 
condition for " Permanent Lines of Bending." 

The use of these lines of bending in simplifying the theory of surfaces is 
the only part of the present method which is new, although the investigations 
connected with them naturally led to the employment of other methods which 
had been used by those who have already treated of this subject. A state- 
ment of the principal methods and results of these mathematicians will save 
repetition, and will indicate the different points of view under which the 
subject may present itself. 

The first and most complete memoir on the subject is that of M. Gauss, 
already referred to. 

The method which he employs consists in referring every point of the 
surface to a corresponding point of a sphere whose radius is unity. Normals 
are drawn at the several points of the surface toward the same side of it, 
then lines drawn through the centre of the sphere in the direction of each of 
these normals intersect the surface of the sphere in points corresponding to 
those points of the original surface at which the normals were drawn. 

If any line be drawn on the surface, each of its points will have a 
corresponding point on the sphere, so that there will be a corresponding Hne 
on the sphere. 

If the line on the surface return into itself, so as to enclose a finite area 
of the surface, the corresponding curve on the sphere will enclose an area on 
the sphere, the extent of which will depend on the form of the surface. 

This area on the sphere has been defined by M. Gauss as the measure of 
the "entire curvature" of the area on the surface. This mathematical quantity 
is of great use in the theory of surfaces, for it is the only quantity connected 
with curvature which is capable of being expressed as the sum of all its parts. 

The sum of the entire curvatures of any number of areas is the entire 
curvature of their sum, and the entire curvature of any area depends on the 
form of its boundary only, and is not altered by any change in the form of 
the surface within the boundary line. 

The curvature of the surface may even be discontinuous, so that we may 
speak of the entire curvature of a portion of a polyhedron, and calculate its 
amount. 

If the dimensions of the closed curve be diminished so that it may be 
treated as an element of the surface, the ultimate ratio of the entire curvature 



TRANSFORMATION OF SURFACES BY BENDING. 89 

to the area of the element on the surface is taken as the measure of the 
" specific curvature " at that point of the surface. 

The terms "entire" and "specific" curvature when used in this paper are 
adopted from M. Gauss, although the use of the sphere and the areas on its 
surface formed an essential part of the original design. The use of these terms 
will save much explanation, and supersede several very cumbrous expressions. 

M. Gauss then proceeds to find several analytical expressions for the measure 
of specific curvature at any point of a surface, by the consideration of three 
points very near each other. 

The co-ordinates adopted are first rectangular, x and y, or x, y and z, being 
regarded as independent variables. 

Then the points on the surface are referred to two systems of curves drawn 
on the surface, and their position is defined by the values of two independent 
variables p and q, such that by varying p while q remains constant, we obtain 
the different points of a line of the first system, while p constant and q 
variable defines a line of the second system. 

By means of these variables, points on the surface may be referred to lines 
on the surface itself instead of arbitrary co-ordinates, and the measure of cur- 
vature may be found in terms of p and q when the surface is known. 

In this way it is shewn that the specific curvature at any point is the 
reciprocal of the product of the principal radii of curvature at that point, a 
result of great interest. 

From the condition of bending, that the length of any element of the 
curve must not be altered, it is shewn that the specific curvature at any point 
is not altered by bending. 

The rest of the memoir is occupied with the consideration of particular 
modes of describing the two systems of lines. One case is when the lines of. 
the first system are geodesic, or "shortest" lines having their origin in a point, 
and the second system is drawn so as to cut off equal lengths from the curv^es 
of the first system. 

The angle which the tangent at the origin of a line of the first system 
makes with a fixed line is taken as one of the co-ordinates, and the distance 
of the point measured along that line as the other. 

It is shewn that the two systems intersect at right angles, and a simple 
expression is found for the specific curvature at any point. 

M. Liouville (Journal, Tom. xii.) has adopted a different mode of simpli- 

VOL. I. 22 



90 TRANSFORMATION OF SURFACES BY BENDING. 

tying the problem. He has shewn that on every surface it is possible to find 
two systems of curves intersecting at right angles, such that the length and 
breadth of every element into which the surface is thus divided shall be equal, 
and that an infinite number of such systems may be found. By means of these 
curves he has found a much simpler expression for the specific curvature than 
that given by M. Gauss. 

He has also given, in a note to his edition of Monge, a method of testing 
two given surfaces in order to determine whether they are applicable to one 
another. He first draws on both surfaces lines of equal specific curvature, and 
determines the distance between two corresponding consecutive lines of curvature 
in both surfaces. 

If by assuming the origin properly these distances can be made equal for 
every part of the surface, the two surfaces can be applied to each other. He 
has developed the theorem analytically, of which this is only the geometrical 
interpretation. 

When the lines of equal specific curvature are equidistant throughout their 
whole length, as in the case of surfaces of revolution, the surfaces may be 
applied to one another in an infinite variety of ways. 

When the specific curvature at every point of the surface is positive and 
equal to a^, the surface may be applied to a sphere of radius a, and when the 
specific curvature is negative = —a" it may be applied to the surface of revo- 
lution which cuts at right angles all the spheres of radius a, and whose centres 
are in a straight line. 

M. Bertrand has given in the Xlllth Vol. of Liouville's Journal a very 
simple and elegant proof of the theorem of M. Gauss about the product of 
the radii of curvature. 

He supposes one extremity of an inextensible thread to be fixed at a point 
in a surface, and a closed curve to be described on the surface by the other 
extremity, the thread being stretched all the while. It is evident that the 
length of such a curve cannot be altered by bending the surface. He then 
calculates the length of this curve, considering the length of the thread small, 
and finds that it depends on the product of the principal radii of curvature 
of the surface at the fixed point. His memoir is followed by a note of 
M. Diguet, who deduces the same result from a consideration of the area of 
the same curve ; and by an independent memoir of M. Puiseux, who seems to 
give the same proof at somewhat greater length. 



TRANSFORMATION OF SURFACES BY BENDING. 91 



Note. Since this paper was written, I have seen the Rev. Professor Jellett's Memoir, On 
the Properties of Inextensible Surfaces. It is to be found in the Transactions of the Royal Irish 
Academy, Vol. XXII. Science, &c., and was read May 23, 18.53. 

Professor Jellett has obtained a system of three partial differential equations which express 
the conditions to which the displacements of a continuous inextensible membrane are subject. 
From these he has deduced the two theorems of Gauss, relating to the invariability of the product 
of the radii of curvature at any point, and of the " entire curvature" of a finite portion of the 
surface. 

He has then applied his method to the consideration of cases in which the flexibihty of the 
surface is limited by certain conditions, and he has obtained the following results : — 

If the displacements of an inextensible surface he all parallel to the same plane, the mrface 
moves as a rigid body. 

Or, more generally, 

If the movement of an inextensible surface, parallel to any one line, be that of a rigid body, the 
entire movement is that of a rigid body. 

The following theorems relate to the case in which a curve traced on the surface is rendered 
rigid :— 

// any curve be traced upon an inextensible surface whose principal radii of curvature are finite 
and of the same sign, and if this curve he rendered immoveable, the entire surface will become 
immoveable also. 

In a developable surface composed of an inextensible membrane, any one of its rectilinear 
sections may be fixed without destroying the fiexibility of the membrane. 

In convexo-concave surfaces, there are two directions passing through every point of the 
surface, such that the curvature of a normal section taken in these directions vanishes. We 
may therefore conceive the entire surface to be crossed by two series of curves, such that 
a tangent drawn to either of them at any point shall coincide with one of these direc- 
tions. These curves Professor Jellett has denominated Curves of Flexure, from the following 
properties : — 

Any curve of fiexure may he fi^ed without destroying the fiexibility of the surface. 

If an arc of a curve traced upon an inextensible surface be rendered fixed or rigid, the entire of 
the quadrilateral, formed by drauring the two curves of fiexure through each extremity of the curve, 
become fixed or rigid also. 

Professor Jellett has also investigated the properties of partially inextensible surfaces, and 
of thin material laminae whose extensibility is small, and in a note he has demonstrated the 
following theorem : — 

If a closed oval surface he perfectly inextensible, it is also perfectly rigid. 

A demonstration of one of Professor Jellett's theorems will be found at the end of this paper. 

J. C. M. 
Aug. 30, 1851 




92 TRANSFORMATION OF SURFACES BY BENDING. 



On the properties of a Surface considered as the limit of the inscribed 

Polyhedron. 

1. To inscribe a polyhedron in a given surface, aU whose sides shall he 
triangles, and all whose solid angles shall he hexahedral. 

On the given surface describe a series of curves 
according to any assumed law. Describe a. second series 
intersecting these in any manner, so as to divide the 
whole surface into quadrilaterals. Lastly, describe a 
third series (the dotted lines in the figure), so as to 
pass through all the intersections of the first and second 
series, forming the diagonals of the quadrilaterals. 

The surface is now covered with a network of curvilinear triangles. The 
plane triangles which have the same angular points will form a polyhedron 
fulfilling the required conditions. By increasing the number of the curves in 
each series, and diminishing their distance, we may make the polyhedron 
approximate to the surface without limit. At the same time the polygons 
formed by the edges of the polyhedron will approximate to the three systems 
of intersecting curves. 

2. To find the measure of the ''entire curvature" of a solid angle of the 
'polyhedron, and of a finite portion of its surface. 

From the centre of a sphere whose radius is unity draw perpendiculars to 
the planes of the six sides forming the solid angle. These lines will meet the 
surface in six points on the same side of the centre, which being joined by 
arcs of great circles will form a hexagon on the surface of the sphere. 

The area of this hexagon represents the entire curvature of the solid angle. 

It is plain by spherical geometry that the angles of this hexagon are the 
supplements of the six plane angles which form the solid angle, and that the 
arcs forming the sides are the supplements of those subtended by the angles 
of the six edges formed by adjacent sides. 

The area of the hexagon is equal to the excess of the sum of its angles 
above eight right angles, or to the defect of the sum of the six plane angles 
from four right angles, which is the same thing. Since these angles are 



TRANSFORMATION OF SURFACES BY BENDING. 93 

invariable, the bending of the polyhedron cannot alter the measure of curvature 
of each of its solid angles. 

If perpendiculars be drawn to the sides of the polyhedron which contain 
other solid angles, additional points on the sphere will be found, and if these 
be joined by arcs of great circles, a network of hexagons will be formed on 
the sphere, each of which corresponds to a solid angle of the polyhedron and 
represents its " entire curvature." 

The entire curvature of any assigned portion of the polyhedron is the sum 
of the entire curvatures of the solid angles it contains. It is therefore repre- 
sented by a polygon on the sphere, which is composed of all the hexagons 
corresponding to its solid angles. 

If a polygon composed of the edges of the polyhedron be taken as the 
boundary of the assigned portion, the sum of its exterior angles will be the 
same as the sum of the exterior angles of the polygon on the sphere ; but 
the area of a spherical polygon is equal to the defect of the sum of its 
exterior angles from four right angles, and this is the measure of entire curva- 
ture. 

Therefore the entire curvature of the portion of the polyhedron enclosed 
by the polygon is equal to the defect of the sum of its exterior angles from 
four right angles. 

Since the entire curvature of each solid angle is unaltered by bending, 
that of a finite portion of the surface must be also invariable. 

3. On the " Conic of Contact," and its use in determining the curvature 
of normal sections of a surface. 

Suppose the plane of one of the triangular facets of the polyhedron to 
be produced till it cuts the surface. The form of the curve of intersection 
\7ill depend on the nature of the surface, and when the size of the triangle 
is indefinitely diminished, it will approximate, to the form of a conic section. 

For we may suppose a surface of the second order constructed so as to 
have a contact of the second order with the given surface at a point within 
the angular points of the triangle. The curve of intersection with this surface 
will be the conic section to which the other curve of intersection approaches. 
This curve will be henceforth called the " Conic of Contact," for want of a better 
name. 



1)4 



TRANSFORMATION OF SURFACES BY BENDING. 



To Jind tJie radius of curvature of a normal section 
of the surface. 

Let ARa be the conic of contact, C its centre, and 
CP perpendicular to its plane. rPR a normal section, and 
its centre of curvature, then 

= 1.^ in the limit, when CR and PR coincide, 
^ CP 

-s CP' 
or calling CP the "sa,gitta," we have this theorem: 

"The radius of curvature of a normal section is equal to the square of 
the corresponding diameter of the conic of contact divided by eight times the 
sagitta." 




4. To insciihe a polyhedron in a given surface, all ivhose sides shcdl he 
plane quadrilaterals, and all whose solid angles shall he tetraliedral. 

Suppose the three systems of curves drawn as described in sect. (1), then 
each of the quadrilaterals formed by the intersection of the first and second 
systems is divided into two triangles by the third system. If the planes of 
these two triangles coincide, they form a plane quadrilateral, and if every such 
pair of triangles coincide, the polyhedron will satisfy the required condition. 

Let ahc be one of these triangles, and acd the 
other, which is to be in the same plane with ahc. 
Then if the plane of ahc be produced to meet the 
surface in the conic of contact, the curve will pass 
through ahc and d. Hence ahcd must be a quad- 
rilateral inscribed in the conic of contact. 

But since ah and dc belong to the same system of curves, they will be 
ultimately parallel when the size of the facets is diminished, and for a similar 
reason, ad and ho will be ultimately parallel. Hence ahcd will become a paral- 
lelogram, but the sides of a parallelogram inscribed in a conic are parallel to 
conjugate diameters. 




TRANSFORMATION OF SURFACES BY BENDING. ©5 

Therefore the directions of two curves of the first and second system at 
their point of intersection must be parallel to two conjugate diameters of the 
conic of contact at that point in order that such a polyhedron may be inscribed. 

Systems of curves intersecting in this manner will be referred to as "conju- 
gate systems." 

5. On the elementary conditions of the applicahilitij of two surfaces. 

It is evident, that if one surface is capable of being appUed to another by 
bending, every point, line, or angle in the first has its corresponding point, line, 
or angle in the second. 

If the transformation of the surface be eflfected without the extension or 
contraction of any part, no line drawn on the surface can experience any change 
in its length, and if this condition be fulfilled, there can be no extension or 
contraction. 

Therefore the condition of bending is, that if any line whatever be drawn 
on the first surface, the corresponding curve on the second surface is equal to it 
in length. All other conditions of bending may be deduced from this. 

6. If two curves on the first surface intersect, the corresponcling curves on the 
second surface intersect at the same angle. 

On the first surface draw any curve, so as to form a triangle with the 
curves already drawn, and let the sides of this triangle be indefinitely dimin- 
ished, by making the new curve approach to the intersection of the former 
curves. Let the same thing be done on the second surface. We shall then 
have two corresponding triangles whose sides are equal each to each, by (5), 
and since their sides are indefinitely small, we may regard them as straight 
lines. Therefore by Euclid i. 8, the angle of the first triangle formed by the 
intersection of the two curves is equal to the corresponding angle of the second. 

7. At any given point of the first surface, two directions can he found, which 
are conjugate to each other with respect to the conic of contact at that point, and 
continue to he conjugate to each other when tJie first surface is transformed into the 
second. 

For let the first surface be transferred, without changing its form, to a 
position such that the given point coincides with the corresponding point of the 
second surface, and the normal to the first surface coincides with that of the 



96 



TRANSFORMATION OF SURFACES BY BENDING. 



second at the same point. Then let the first surface be turned about the normal 
as an axis till the tangent of any line through the point coincides with the 
tangent of the corresponding line in the second surface. 

Then by (6) any pair of corresponding lines passing through the point will 
have a common tangent, and will therefore coincide in direction at that point. 

If we now draw the conies of contact belonging to each surface we shall 
have two conies with the same centre, and the problem is to determine a pair 
of conjugate diameters of the first which coincide with a pair of conjugate 
diameters of the second. The analytical solution gives two directions, real, 
coincident, or impossible, for the diameters required. 

In our investigations we can be concerned only with the case in which these 
directions are real. 

When the conies intersect in four points, P, Q, R, S, FQES is a parallelo- 
gram inscribed in both conies, and the axes CA, CB, 
parallel to the sides, are conjugate in both conies. 

If the conies do not intersect, describe, through any 
point P of the second conic, a conic similar to and con- 
centric with the first. If the conies intersect in four 
points, we must proceed as before; if they touch in two 
points, the diameter through those points and its conju- 
gate must be taken. If they intersect in two points only, 
then the problem is impossible ; and if they coincide 
altogether, the conies are similar and similarly situated, 
and the problem is indeterminate. 




8. Two surfaces being given as before, one pair of conjugate systems of 
curves may be drawn on the first surface, which shall correspond to a pair of 
conjugate systems on the second surface. 

By article (7) we may find at every point of the first surface two 
directions conjugate to one another, corresponding to two conjugate directions on 
the second surface. These directions indicate the directions of the two systems 
of curves which pass through that point. 

Knowing the direction which every curve of each system must have at every 
point of its course, the systems of curves may be either drawn by some direct 
geometrical method, or constructed from their equations, which may be found by 
solving their difierential equations. 



TRANSFORMATION OF SURFACES BY BENDING. 97 

Two systems of curves being drawn on the first surface, the corresponding 
systems may be drawn on the second surface. These systems being conjugate 
to each other, fulfil the condition of Art. (4), and may therefore be made the 
means of constructing a polyhedron with quadrilateral facets, by the bending of 
which the transformation may be effected. 

These systems of curves will be referred to as the "first and second systems 
of Lines of Bending." 

9. General considerations applicable to Lines of Bending. 

It has been shewn that when two forms of a surface are given, one of 
which may be transformed into the other by bending, the nature of the Hnes 
of bending is completely determined. Supposing the problem reduced to its 
analyticid expression, the equations of these curves would appear under the 
form of double solutions of differential equations of the first order and second 
degree, each of which would involve one arbitrary quantity, by the variation of 
which we should pass from one curve to another of the same system. 

Hence the position of any curve of either system depends on the value 
assumed for the arbitrary constant ; to distinguish the systems, let us call one 
the first system, and the other the second, and let all quantities relating to 
the second system be denoted by accented letters. 

Let the arbitrary constants introduced by integration be u for the first 
system, and u for the second. 

Then the value of lo will determine the position of a curve of the first 
system, and that of u a curve of the second system, and therefore u and u will 
suffice to determine the point of intersection of these two curves. 

Hence we may conceive the position of any point on the surface to be 
determined by the values of u and u for the curves of the two systems which 
intersect at that point. 

By taking into account the equation to the surface, we may suppose x, y, 
and 2 the co-ordinates of any point, to be determined as functions of the two 
variables u and u. This being done, we shall have materials for calculating 
everything connected with the surface, and its lines of bending. But before 
entering on such calculations let us examine the principal properties of these lines 
which we must take into account. 

Suppose a series of values to be given to u and u, and the corresponding 
curves to be drawn on the surface. 

VOL, I. 13 



98 TRANSFORMATION OF SURFACES BY BENDING. 

The surface will then be covered with a system of quadrilaterals, the size 
of which may be diminished indefinitely by interpolating values of u and u 
between those already assumed; and in the limit each quadrilateral may be 
regarded as a parallelogram coinciding with a facet of the inscribed polyhedron. 

The length, the breadth, and the angle of these parallelograms will vary at 
different parts of the surface, and will therefore depend on the values of u 
and It. 

The curvature of a line drawn on a surface may be investigated by consider- 
ing the curvature of two other lines depending on it. 

The first is the projection of the line on a tangent plane to the surface at 
a given point in the line. The curvature of the projection at the point of 
contact may be called the tangential cwvature of the line on the surface. It 
has also been called the geodesic curvature, because it is the measure of its 
deviation from a geodesic or shortest line on the surface. 

The other projection necessary to define the curvature of a line on the 
surface is on a plane passing through the tangent to the curve and the normal 
to the surface at the point of contact. The curvature of this projection at that 
point may be called the normal cw^ature of the line on the surface. 

It is easy to shew that this normal curvature is the same as the curvature 
of a normal section of the surface passing through a tangent to the curve at 
the same point. 



10. General considerations applicable to the inscribed polyhedron. 

When two series of lines of bending belonging to the first and second systems 
have been described on the surface, we may proceed, as in Art. (l), to describe 
a third series of curves so as to pass through all their intersections and form 
the diagonals of the quadrilaterals foi-med by the first pair of systems. 

Plane triangles may then be constituted within the surface, having these 
points of intersection for angles, and the size of the facets of this polyhedron may 
be diminished indefinitely by increasing the number of curves in each series. 

But by Art. (8) the first and second systems of lines of bending are conju- 
gate to each other, and therefore by Art. (4) the polygon just constructed will 
have every pair of triangular facets in the same plane, and may therefore be 



TRANSFORMATION OF SURFACES BY BENDING. 99 

considered as a polyhedron with plane quadrilateral facets all whose solid angles 
are formed by four of these facets meeting in a point. 

When the number of curves in each system is increased and their distance 
diminished indefinitely, the plane facets of the polyhedron will ultimately coincide 
with the curved surface, and the polygons formed by the successive edges between 
the facets, will coincide with the lines of bending. 

These quadrilaterals may then be considered as parallelograms, the length 
of which is determined by the portion of a curve of the second system inter- 
cepted between two curves of the first, while the breadth is the distance of 
two curves of the second system measured along a curve of the first. The 
expressions for these quantities will be given when we come to the calculation of 
our results along with the other particulars which we only specify at present. 

The angle of the sides of these parallelograms will be ultimately the same 
as the angle of intersection of the first and second systems, which we may 
call <f> ; but if we suppose the dimensions of the facets to be small quantities 
of the first order, the angles of the four facets which meet in a point will difier 
from the angle of intersection of the curves at that point by small angles of 
the first order depending on the tangential curvature of the lines of bending. 
The sum of these four angles will differ from four right angles by a small 
angle of the second order, the circular measure of which expresses the entire 
curvature of the solid angle as in Art. (2). 

The angle of inclination of two adjacent facets will depend on the normal 
curvature of the lines of bending, and will be that of the projection of two con- 
secutive sides of the polygon of one system on a plane perpendicular to a side 
of the other system. 

11. Explanation of the Notation to be employed in calculation. 

Suppose each system of lines of bend- 
ing to be determined by an equation con- 
taining one arbitrary parameter. 

Let this parameter be u for the first 
system, and u' for the second. 

Let two curves, one from each system, 
be selected as curves of reference, and let 
their parameters be u^ and u\. 




100 TRANSFORMATION OF SURFACES* BY BENDING. 

Let ON and OM in the figure represent these two curves. 

Let PM be any curve of the first system whose parameter is u, and PN 
any curve of the second whose parameter is u, then their intersection P may 
be defined as the point (w, u'), and all quantities referring to the point P may 
be expressed as functions of u and u. 

Let PN, the length of a curve of the second system (u), from N (wj to P 
(u), be expressed by s, and PM the length of the curve {u) from {u\) to (u), by 
s\ then s and s will be functions of u and u. 

Let (w + Sm) be the parameter of the curve QF of the first system consecu- 
tive to PM. Then the length of PQ, the part of the curve of the second system 
intercepted between the curves (u) and (w + Sw), will be 

ds ^ 
du 

Similarly PR may be expressed by 

ds\ , 

These values of PQ and PR will be the ultimate values of the length and 
breadth of a quadrilateral facet. 

The angle between these lines will be ultimately equal to ^, the angle of 
intersection of the system ; but when the values of 8w and hu are considered as 
finite though small, the angles a, 6, c, d of the facets which form a soHd angle 
will depend on the tangential curvature of the two systems of lines. 

Let T be the tangential curvature of a curve of the first system at the 
given point measured in the direction in which u increases, and let r\ that of the 
second system, be measured in the direction in which xC increases. 

Then we shall have for the values of the four plane angles which meet at P, 

, \ ds ^ , 1 ds^ 

1 _, 1 c?/ ^ . 1 ds ^ 

~^ It du It du ' 

, \ ds rs , \ ds ^ 
J . I ds' , 1 ds ^ 



TRANSFORMATION OF SURFACES BY BENDING. 101 

These values are correct as far as the first order of small quantities. Those 
corrections which depend on the curvature of the surface are of the second order. 

Let p be the normal curvature of a curve of the first system, and p that 
of a curve of the second, then the inclination I of the plane facets a and 6, 
separated by a curve of the second system, will be 

p sin ^ du 
as far as the first order of small angles, and the inclination V of h and c will be 



7/ 1 0^ ^ 

/ = -7—. — 7 -J- ou 
p Bin.<f> du 



to the same order of exactness. 




12. On the corresponding polygon on the surface of the sphere of reference. 

By the method described in Art. (2) we may 
find a point on the sphere corresponding to each 
facet of the polyhedron. 

In the annexed figure, let a, b, c, d be the 
points on the sphere corresponding to the four facets 
which meet at the solid angle P. Then the area 
of the spherical quadrilateral a, h, c, d will be the 
measure of the entire curvature of the solid angle P. 

This area is measured by the defect of the sum of the exterior angles 
from four right angles ; but these exterior angles are equal to the four angles 
a, h, c, d, which form the solid angle P, therefore the entire curvature is 
measured by 

k = 2'rr-{a + h + c-{-d). 

Since a, h, c, d are invariable, it is evident, as in Art. (2), that the entire 
curvature at P is not altered by bending. 

By the last article it appears that when the facets are small the angles b 
and d are approximately equal to <j), and a and c to (tt — ^), and since the sides 
of the quadrilateral on the sphere are small, we may regard it as approximately 
a plane parallelogram whose angle bad = <f). 

The sides of this parallelogram will be I and I', the supplements of the 
angles of the edges of the polyhedron, and we may therefore express its area 
as a plane parallelogram 

k = IV sin <f>. 



102 



TRANSFORMATION OF SURFACES BY BENDING. 



By the expression for I and V in the last article, we find 

, 1 ds ds\ ^ , 

k = — r-. — 7 J- J-/ ou du 
pp sm<^ du du 

for the entire curvature of one solid angle. 

Since the whole number of solid angles is equal to the whole number of 
facets, we may suppose a quarter of each of the facets of which it is composed 
to be assigned to each solid angle. The area of these will be the same as that 

of one whole facet, namely, 

, ds ds' o ^ , 
sm 9 -J- T-> ou ou ; 

therefore dividing the expression for k by this quantity, we find for the value 

of the specific curvature at P 

1 
■^ pp sm'<^ 
which gives the specific curvature in terms of the normal curvatures of the 
lines of bending and their angle of intersection. 

13. Further reduction of this expression by rmans of the " Conic of Con- 
tact" as defined in Art. (3). 

Let a and b be the semiaxes of the conic of contact, and h the sagitta 
or perpendicular to its plane from the centre to the surface. 

Let CP, CQ be semidiameters parallel to the 
lines of bending of the first and second systems, and 
therefore conjugate to each other. 



By (Art. 3), 



, CP" 

p=^-hr 




and p=i-j^; 
and the expression for p in Art. (12), becomes 

^~{CP.CQsm(t>)'' 

But CP .CQbukJ) is the area of the parallelogram CPRQ, which is one 

quarter of the circumscribed parallelogram, and therefore by a well-known 

theorem 

CP .CQsm4> = ah, 



TRANSFORMATION OF SURFACES BY BENDING. 103 

and the expression for p becomes 

or if the area of the circumscribing parallelogram be called A, 

The principal radii of curvature of the surface are parallel to the axes of 
the conic of contact. Let H and i^ denote these radii, then 

and therefore substituting in the expression for p, 

1 

or the specific curvature is the reciprocal of the product of the principal radii 
of curvature. 

This remarkable expression was introduced by Gauss in the memoir referred 
to in a former part of this paper. His method of investigation, though not 
80 elementary, is more direct than that here given, and wUl shew how this 
result can be obtained without reference to the geometrical methods necessary 
to a more extended inquiry into the modes of bending. 

14. 0)1 the variation of normal curvature of the lines of bending as we pa^s 
from one point of the surface to another. 

We have determined the relation between the normal curvatures of the 
lines of bending of the two systems at their points of intersection; we have 
now to find the variation of normal curvature when we pass from one hne of 
the first system to another, along a line of the second. 

In analytical language we have to find the value of 

du \pj 

Referring to the figure in Art. (11), we shall see that this may be done 
if we can determine the difierence between the angle of inclination of the 
facets a and h, and that of c and d : for the angle I between a and b is 

J 1 ds 5. , 
psiJKp du 



104 TRANSFORMATION OF SURFACES BY BENDING. 

and therefore the difference between the angle of a and b and that of c and d is 

~ du ~ du \psm<f> du'j 
whence the differential of p with respect to u may be found 

We must therefore find U, and this is done by means of the quadrilateral 
on the sphere described in Art. (12). 




15. To find the values of hi and U\ 

In the annexed figure let ahcd repre- 
sent the small quadrilateral on the surface 
of the sphere. The exterior angles a, h, 
c, d are equal to those of the four facets 
which meet at the point P of the surface, 
and the sides represent the angles which 
the planes of those facets make with each 
other ; so that 

ah = l, lc = l\ cd = l + U, da = l' + Br, 

and the problem is to determine Bl and hi" in terms of the sides I and V and 
the angles a, h, c, d. 

On the sides ha, he complete the parallelogram ahcd. 

Produce ad to p, so that ap = aS. Join Bp. 
Make eq = cd and join dq. 
then Bl = cd- ah, 
= cq — ch, 
= -(qo + oB), 

Now qo = qd tan qdo 

= cd sin qcd cot qod, 
but cd = I nearly, sin qcd = qcd==(e + h-7r) and qod = <f>; 
.'. qo^l (c + h- it) cot <f>. 



TRANSFORMATION OF SURFACES BY BENDING. 105 

Also oS = -—-^ — 
Sin bop 

= aB (Bap) — — 7 
^ ^' 8m<f> 

= l'(a+h-7T)J-r. 

Substituting the values of a, h, c, d from Art. (11), 
Sl= — (qo + 08) 

= —I —, ^- cot <i>Su — V — T—, - — r Bu. 
r du ^ r du sm0 

Finally, substituting the values of I, V, and Bl from Art. (14), 

d ( \ ds"\ sj 5 , cot (/) cZs' 1 (i5 5. ^ , 1 ds I ds' ^ , 

du \p sin <p du / p sm <f> du r du p sm <j> du r du 

which may be put under the more convenient form 

— n ^ = — 1 / 1 ^^'\ 1 ds , p I ds 1 
du^ °'^'~du ^ \sin <j> du) r du ^ p' r du sin <^ ' 

and from the value of Bl' we may similarly obtain 

d ,, '\ _ _^ 1 / 1 ^\ ,i^ +^j_^i^ ^ 
du ^ ^ ^ ' du' ° \sin <f> du) r du' ^ p r du sin (ft ' 

We may simplify these equations by putting p for the specific curvature of 

the surface, and q for the ratio , , which is the only quantity altered by bending. 

We have then 

p = — / . , . , and q = —,, 
^ pp sm=<^' ^ p 

whence p' = q — ^^-r , p'^ = t-tj y 

^ ^ p sin <f) 9. P s^ Y 

and the equations become 

d ,. \ d , ( ^Tl'X 1 ds , , 2 ds 1 

In this way we may reduce the problem of bending a surface to the 
consideration of one variable q, by means of the lines of bending. 

VOL. I. 14 



d_ 

du' 



106 TRANSFORMATION OF SURFACES BY BENDING. 

16. To obtain the conditio of Instantaneous lines of bending. 

We have now obtained tlie values of the differential coefficients of q with 
respect to each of the variables u, u. 
From the equation 

we might find an equation which would give certain conditions of lines of 
bending. These conditions however would be equivalent to those which we have 
already assumed when we drew the systems of lines so as to be conjugate to 
each other. 

To find the true conditions of bending we must suppose the form of the 
surface to vary continuously, so as to depend on some variable t which we 
may call the time. 

Of the difierent quantities which enter into our equations, none are changed 
by the operation of bending except q, so that in differentiating with respect 
to t all the rest may be considered constant, q being the only variable. 

Differentiating the equations of last article with respect to t, we obtain 
d" ,, . 2 ds 1 d ,, . 



Whence 



c?" ,, . 2 ds' 1 I d ,. . 



A^t'^^'^^^ = 



{.4 1- 1 si^)-'^ Tu ^, ii'^^H^'o^'^' 1 1 ii^^ 3^.<(">^*)- 



and 



(log l) 



dududt 



( d /2ds 1 \ 2 ds 1 d , } ^ d ,, 2 ds 1 1 d ,, . 

{M?d^^^'r-di7^^d^^'^'irqdt^^'^^^ 

two independent values of the same quantity, whence the requiied conditions 
may be obtained. 



TRANSFORMATION OF SURFACES BY BENDING. 



107 



Substituting in these equations the values of those quantities which occur 
in the original equations, we obtain 



I ds ( d , , 



ds 
du 



sin 



*) 



+ - , \, cot <!> y 



2 ds 
r du 



\l ds ( d , f ,ds . A 2 ds . ,\ 



which is the condition which must hold at every instant during the process of 
bending for the lines about which the bending takes place at that instant. 
When the bending is such that the position of the lines of bending on the 
surface alters at every instant, this is the only condition which is required. 
It is therefore called the condition of Instantaneous lines of bending. 

17. To find the condition of Permanent lines of bending. 

Since q changes with the time, the equation of last article will not be 
satisfied for any finite time unless both sides are separately equal to zero. In 
that case we have the two conditions 



(!)■ 



d , / ds . ,\ 2ds ^ , ^^ 
^,log(i^r^^sm<^j + -^,cotc^ = 0, 


y 


1 ds ^ 
or - -J- = 0. 
r du 

|^log(i>r'^,siD<^)+|^cot<^ = 0,' 




1 d/ ^ 
or -, -J-, = 0. 
r du 





(2). 



If the lines of bending satisfy these conditions, a finite amount of bending 
may take place without changing the position of the system on the surface. 
Such lines are therefore called Permanent lines of bending. 

The only case in which the phenomena of bending may be exhibited by 
means of the polyhedron with quadrilateral facets is that in which permanent 
lines of bending are chosen as the boundaries of the facets. In all other cases 
the bending takes place about an instantaneous system of lines which is con- 
tinually in motion with respect to the surface, so that the nature of the poly- 
hedron would need to be altered at every instant. 



14—2 



108 TRANSFORMATION OF SURFACES BY BENDING. 

We are now able to determine whether any system of lines drawn on a 
given surface is a system of instantaneous or permanent lines of bending. 

We are also able, by the method of Article (8), to deduce from two con- 
secutive forms of a surface, the lines of bending about which the transformation 
must have taken place. 

If our analytical methods were sufficiently powerful, we might apply our 
results to the determination of such systems of lines on any known surface, but 
the necessary calculations even in the simplest cases are so compHcated, that, 
even if useful results were obtained, they would be out of place in a paper of 
this kind, which is intended to afford the means of forming distinct conceptions 
rather than to exhibit the results of mathematical labour. 

18. On the application of the ordinary unethods of analytical geometry to the 
consideration of lines of bending. 

It may be interesting to those who may hesitate to accept results derived 
from the consideration of a polyhedron, when applied to a curved surface, to 
inquire whether the same results may not be obtained by some independent 
method. 

As the following method involves only those operations which are most 
familiar to the analyst, it will be sufficient to give the rough outline, which may 
be filled up at pleasure. 

The proof of the invariability of the specific curvature may be taken from 
any of the memoirs above referred to, and its value in terms of the equation of 
the surface will be foimd in the memoir of Gauss. 

Let the equation to the surface be put under the form 

then the value of the specific curvature is 



d\ dh d^ 



dot? dif dx 



~dJz'^ dz^ 
dx dy\ 



The definition of conjugate systems of curves may be rendered independent 
of the reasoning formerly employed by the following modification. 



TRANSFORMATION OF SURFACES BY BENDINO. 109 

Let a tangent plane move along any line of the first system, then if the line 
of ultimate intersection of this plane with itself be always a tangent to some line 
of the second system, the second system is said to be conjugate to the first. 

It is easy to show that the first system is also conjugate to the second. 

Let the system of curves be projected on the plane of xy, and at the point 
(x, y) let a be the angle which a projected curve of the first system makes with 
the axis of x, and /8 the angle which the projected curve of the second system 
which intersects it at that point makes with the same axis. Then the condition 
of the systems being conjugate will be found to be 

a and y3 being known as functions of x and y, we may determine the nature 
of the curves projected on the plane of xy. 

Supposing the surface to touch that plane at the origin, the length and 
tangential curvature of the lines on the surface near the point of contact may 
be taken the same as those of their projections on the plane, and any change 
of form of the surface due to bending will not alter the form of the projected 
lines indefinitely near the point of contact. We may therefore consider z as the 
only variable altered by bending; but in order to apply our analysis with facility, 
we may assume 

72 

^ = Pg sin' a + PQ- sin' A 

d'z 

, J = — PQ sin a cos a — PQ~^ sin y3 cos ^, 

^ = PQ cos' a + P^-^ cos' /8. 

It will be seen that these values satisfy the condition last given. Near the 
origin we have 



d*z dh d\ I* n- . , / n\ 



and q=Q'*. 



110 TRANSFORMATION OF SURFACES BY BENDING. 

Differentiating these values of -y-^ , &c., we shall obtain two values of , , 
and of 1—7—3, which being equated will give two equations of condition. 

Now if s' be measured along a curve of the first system, and R be any 

function of x and y, then 

dE dR dR . 

-^j-y = -^j- cos a + -7- sm a, 
as dx ay 

, dR _ dR ds' 
du' ds du ' 

We may also show that -=-^ = - , 



, ,, , da . da d . (ds' . ,\ 

and that cos a ;i — sm a ;t- = t- log ( -j—, sm 1 . 
cty (j/X cLs \ci/U I 



By substituting these values in the equations thus obtained, they are 
reduced to the two equations given at the end of (Art. 15). This method of 
investigation introduces no difficulty except that of somewhat long equations, and 
is therefore satisfactory as supplementary to the geometrical method given at 
length. 

As an example of the method given in page (2), we may apply it to 
the case of the surface whose equation is 



(^.) *{rf-j-©' 



This surface may be generated by the motion of a straight line whose 
equation is of the form 



= acosnl — j, 2/ = asinni-f- 



t being the variable, by the change of which we pass from one position of the 
line to another. This line always passes through the circle 

z = 0, ar' + y = a', 
and the straight lines z = c, cc=^0, 
and z— —c, y = 0, 
which may therefore be taken as the directors of the surface. 



TRANSFORMATION OF SURFACES BY BENDING. Ill 

Taking two consecutive positions of this line, in which the values of t 
are t and t + Bt, we may find by the ordinary methods the equation to the 
shortest line between them, its length, and the co-ordinates of the point in which 
it intersects the first line. 



Calling the length 8^, 



ac 



8C= ,/^ Bin 2tBt, 



Ja' + c 
and the co-ordinates of the point of intersection are 

x = 2a cos' t, y = 2a sin* t, z= —c cos 2t. 
The angle 80 between the consecutive lines is 

Ja- + c 
The distance So- between consecutive shortest lines is 

^ 3a'-F-2c* 



and the angle S<^ between these latter lines is 



sin 2t8t, 



'Ja' + c 

Hence if we suppose ^, 6, cr, (f), and t to vanish together, we shall have by 
integration 

(T = ~—, ( 1 — cos 2t), 

Ja' + c' 

By bending the surface about its generating lines we alter the value of (ft 
in any manner without changing 4, 0, or or. For instance, making <^ = 0, all the 
generating lines become parallel to the same plane. Let this plane be that of 
xy, then ^ is the distance of a generating line from that plane. The projections 



o- = 



112 TRANSFORMATION OF SURFACES BY BENDING. 

of the generating lines on the plane of xy will, by their ultimate intersections, 
form a curve, the length of which is measured by a, and the angle which its 
tangent makes with the axis of x hj 0, 6 and o- being connected by the equation 

^ I 1 - cos 6 , 

which shows the curve to be an epicycloid. 

The generating lines of the surface when bent into this form are therefore 
tangents to a cylindrical surface on an epicycloidal base, touching that surface 
along a curve which is always equally inclined to the plane of the base, the 
tangents themselves being drawn parallel to the base. 

We may now consider the bending of the surface of revolution 

Putting r = Jaf + f, then the equation of the generating line is 

r^ + z^ = c^. 
This is the well-known hypocycloid of four cusps. 

Let s be the length of the curve measured from the cusp in the axis of z, 
then, 

s = |<jV\ 
wherefore, r = (|)' c " * 5^. 

Let 6 be the angle which the plane of any generating line makes with 
that of xz, then s and 6 determine the position of any point on the surface. 
The length and breadth of an element of the surface will be Ss and rB$. 

Now let the surface be bent in the manner formerly described, so that 
becomes 0^, and r, r, when 

0^ = 1x0 and r' = -ry 
then r' = (f)'c-V"'s' 

provided o' = /u,'c. 
The equation between r' and s being of the same form as that between 
r and ^ shows that the surface when bent is similar to the original surface, its 
dimensions being multiphed by fi*. 



TRANSFORMATION OF SURFACES BY BENDING. 113 

This, however, is true only for one half of the surface when bent. The 
other half is precisely symmetrical, but belongs to a surface which is not con- 
tinuous with the first. 

The surface in its original form is divided by the plane of xy into two 
parts which meet in that plane, forming a kind of cuspidal edge of a circular 
form which limits the possible value of s and r. 

After being bent, the surface still consists of the same two parts, but the 
edge in which they meet is no longer of the cuspidal form, but has a finite 

angle = 2 cos"^ - , and the two sheets of the surface become parts of two different 

surfaces which meet but are not continuous. 



NOTE. 

As an example of the application of the more general theory of " lines of bending," let us 
consider the problem which has been already solved by Professor Jellett. 

To determine the conditions under which one portion of a surface may he rendered rigid, while 
the remainder is flexible. 

Suppose the lines of bending to be traced on the surface, and the corresponding poly- 
hedron to be formed, as in (9) and (10), then if the angle of one of the four edges which 
meet at any solid angle of the polyhedron be altered by bending, those of the other three 
must be also altered. These edges terminate in other solid angles, the forms of which will 
also be changed, and therefore the efifect of the alteration of one angle of the polyhedron will 
be communicated to every other angle within the system of lines of bending which defines 
the form of the polyhedron. 

If any portion of the surface remains unaltered it must lie beyond the limits of the 
system of lines of bending. We must therefore investigate the conditions of such a system 
being bounded. 

The boundary of any system of lines on a surface is the curve formed by the ultimate inter- 
section of those lines, and therefore at any given point coincides in direction with the curve of 
the system which passes through that point. In this case there are two systems of lines of 
bending, which are necessarily coincident in extent, and must therefore have the same boundary. 
At any point of this boundary therefore the directions of the lines of bending of the first 
and second systems are coincident. 

But, by (7), these two directions must be "conjugate" to each other, that is, must corre- 
spond to conjugate diameters of the "Conic of Contact." Now the only case in which con- 
VOL. I. 15 



114 TRANSFORMATION OF SURFACES BY BENDING. 

jugate diameters of a conic can coincide, is when the conic is an hyperbola, and both diameters 
coincide with one of the asymptotes ; therefore the boundary of the system of lines of bending 
must be a curve at every point of which the conic of contact is an hyperbola, one of whose 
asymptotes lies in the direction of the curve. The radius of " normal curvature " must there- 
fore by (3) be infinite at eveiy point of the curve. This is the geometrical property of 
what Professor Jellett calls a " Curve of Flexure," so that we may express the result as 
follows : 

If one portion of a surface be fixed, while the remainder is bent, the boundary of the fixed 
portion is a curve of fiexure. 

This theorem includes those given at p. (92), relative to a fixed curve on a surface, for in 
a surface whose curvatures are of the same sign, there can be no "curves of flexure," and 
in a developable surface, they are the rectilinear sections. Although the cuspidal edge, or 
arete de rebroussement, satisfies the analytical condition of a curve of flexure, yet, since its 
form determines that of the whole surface, it cannot remain fixed while the form of the surface 
is changed. 

In concavo-convex surfaces, the curves of flexure must either have tangential curvature or 
be straight lines. Now if we put <^=0 in the equations of Art. (17), we find that the 
lines of bending of both systems have no tangential curvature at the point where they touch 
the curve of flexure. They must therefore lie entirely on the convex side of that curve, and 
therefore 

If a curve of fiexure be fi^ed, the surface on the concave side of the curve is not flexible. 

I have not yet been able to determine whether the surface is inflexible on the convex side 
of the curve. It certainly is so in some cases which I have been able to work out, but I 
have no general proof. 

When a surface has one or more rectilinear sections, the portions of the surface between 
them may revolve as rigid bodies round those lines as axes in any manner, but no other motion 
is possible. The case in which the rectilinear sections form an infinite series has been discussed 
in Sect. (I.). 



[From the Cambridge and Dublin Mathematical Journal, Vol. ix. 



V. On a particular case of the descent of a heavy body in a resisting 

medium. 

Every one must have observed that when a slip of paper falls through 
the air, its motion, though undecided and wavering at first, sometimes becomes 
regular. Its general path is not in the vertical direction, but inclined to it 
at aji angle which remains nearly constant, and its fluttering appearance will 
be found to be due to a rapid rotation round a horizontal axis. The direction 
of deviation from the vertical depends on the direction of rotation. 

If the positive directions of an axis be toward the right hand and upwards, 
and the positive angular direction opposite to the direction of motion of the 
hands of a watch, then, if the rotation is in the positive direction, the hori- 
zontal part of the mean motion will be positive. 

These efiects are commonly attributed to some accidental peculiarity in the 
form of the paper, but a few experiments with a rectangular slip of paper 
(about two inches long and one broad), will shew that the direction of rotation 
is determined, not by the irregularities of the paper, but by the initial circum- 
stances of projection, and that the symmetry of the form of the paper greatly 
increases the distinctness of the phenomena. We may therefore assume that 
if the form of the body were accurately that of a plane rectangle, the same 
effects would be produced. 

The following investigation is intended as a general explanation of the true 
cause of the phenomenon. 

I suppose the resistance of the air caused by the motion of the plane to 
be in the direction of the normal and to vary as the square of the velocity 
estimated in that direction. 

Now though this may be taken as a sufficiently near approximation to the 
magnitude of the resisting force on the plane taken as a whole, the pressure 

15—2 



116 DESCENT OF A HEAVY BODY IN A RESISTING MEDIUM. 

on any given element of the surface will vary with its position so that the 
resultant force will not generally pass through the centre of gravity. 

It is found by experiment that the position of the centre of pressure 
depends on the tangential part of the motion, that it lies on that side of the 
centre of gravity towards which the tangential motion of the plane is directed, 
and that its distance from that point increases as the tangential velocity in- 
creases. 

I am not aware of any mathematical investigation of this effect. The 
explanation may be deduced from experiment. 

Place a body similar in shape to the sHp of paper obliquely in a current 
of some visible fluid. Call the edge where the fluid first meets the plane the 
first edge, and the edge where it leaves the plane, the second edge, then we 
may observe that 

(1) On the anterior side of the plane the velocity of the fluid increases 
as it moves along the surface from the first to the second edge, and therefore 
by a known law in hydrodynamics, the pressure must diminish from the first 
to the second edge. 

(2) The motion of the fluid behind the plane is very unsteady, but may 
be observed to consist of a series of eddies diminishing in rapidity as they 
pass behind the plane from the first to the second edge, and therefore relieving 
the posterior pressure most at the first edge. 

Both these causes tend to make the total resistance greatest at the first 
edge, and therefore to bring the centre of pressure nearest to that edge. 

Hence the moment of the resistance about the centre of gravity will always 
tend to turn the plane towards a position perpendicular to the direction of the 
current, or, in the case of the slip of paper, to the path of the body itself. It 
will be shewn that it is this moment that maintains the rotatory motion of 
the falling paper. 

When the plane has a motion of rotation, the resistance will be modified 
on account of the unequal velocities of difierent parts of the surface. The 
magnitude of the whole resistance at any instant will not be sensibly altered 
if the velocity of any point due to angular motion be small compared with that 
due to the motion of the centre of gravity. But there will be an additional 
moment of the resistance round the centre of gravity, which will always act in 
the direction opposite to that of rotation, and wOl vary directly as the normal 
and angular velocities together. 



DESCENT OF A HEAVY BODY IN A RESISTING MEDIUM. 117 

The part of the moment due to the obliquity of the motion will remain 
nearly the same as before. 

We are now prepared to give a general explanation of the motion of the 
slip of paper after it has become regular. 

Let the angular position of the paper be determined by the angle between 
the normal to its surface and the axis of x, and let the angular motion be 
such that the normal, at first coinciding with the axis of x, passes towards 
that of y. 

The motion, speaking roughly, is one of descent, that is, in the negative 
direction along the axis of y. 

The resolved part of the resistance in the vertical direction will always 
act upwards, being greatest when the plane of the paper is horizontal, and 
vanishing when it is vertical. 

When the motion has become regular, the effect of this force during a 
whole revolution will be equal and opposite to that of gravity during the same 
time. 

Since the resisting force increases while the normal is in its first and third 
quadrants, and diminishes when it is in its second and fourth, the maxima of 
velocity will occur when the normal is in its first and third quadrants, and 
the minima when it is in the second and fourth. 

The resolved part of the resistance in the horizontal direction will act in 
the positive direction along the axis of x in the first and third quadrants, and 
in the negative direction during the second and fourth; but since the resistance 
increases with the velocity, the whole effect during the first and third quadrants 
will be greater than the whole effect during the second and fourth. Hence 
the horizontal part of the resistance will act on the whole in the positive 
direction, and will therefore cause the general path of the body to incline in 
that direction, that is, toward the right. 

That part of the moment of the resistance about the centre of gravity 
which depends on the angular velocity will vary in magnitude, but wUl always 
act in the negative direction. The other part, which depends on the obliquity 
of the plane of the paper to the direction of motion, will be positive in the 
first and third quadrants and negative in the second and fourth ; but as its 
magnitude increases with the velocity, the positive effect will be greater than 
the negative. 

When the motion has become regular, the effect of this excess in the 



118 DESCENT OF A HEAVY BODY IN A RESISTING MEDIUM. 

positive direction will be equal and opposite to the negative effect due to the 
angular velocity during a whole revolution. 

The motion will then consist of a succession of equal and similar parts 
performed in the same manner, each part corresponding to half a revolution of 
the paper. 

These considerations will serve to explain the lateral motion of the paper, 
and the maintenance of the rotatory motion. 

Similar reasoning will shew that whatever be the initial motion of the 
paper, it cannot remain uniform. 

Any accidental oscillations will increase till their amphtude exceeds half a 
revolution. The motion will then become one of rotation, and will continually 
approximate to that which we have just considered. 

It may be also shewn that this motion will be unstable unless it take 
place about the longer axis of the rectangle. 

If this axis is incHned to the horizon, or if one end of the slip of paper 
be different from the other, the path will not be straight, but in the form of 
a helix. There will be no other essential difference between this case and that 
of the symmetrical arrangement. 

Trinity College, April 5, 1853. 



[From the Transactions of the Royal Scottish Society of Arts, Vol. iv. Part in] 



VI. On the Theory of Colours in relation to Colour-Blindness. 
A letter to Dr G. Wilson. 

Dear Sir, — As you seemed to think that the results which I have obtained 
in the theory of colours might be of service to you, I have endeavoured to 
arrange them for you in a more convenient form than that in which I first 
obtained them. I must premise, that the first distinct statement of the theory 
of colour which I adopt, is to be found in Young's Lectures on Natural Philo- 
sophy (p. 345, Kelland's Edition) ; and the most philosophical enquiry into it 
which I have seen is that of Helmholtz, which may be found in the Annals of 
Philosophy for 1852. 

It is well known that a ray of light, from any source, may be divided by 
means of a prism into a number of rays of different refranglbility, forming a 
series called a spectrum. The intensity of the light is different at different 
points of this spectrum ; and the law of intensity for different refrangibilities 
differs according to the nature of the incident light. In Sir John F. W. 
Herschel's Treatise on Light, diagrams will be found, each of which represents 
completely, by means of a curve, the law of the intensity and refranglbility of 
a beam of solar light after passing through -various coloured media. 

I have mentioned this mode of defining and registering a beam of light, 
because it is the perfect expression of what a beam of light is in itself, con- 
sidered with respect to all its properties as ascertained by the most refined 
instruments. When a beam of light falls on the human eye, certain sensations 
are produced, from which the possessor of that organ judges of the colour and 
intensity of the light. Now, though every one experiences these sensations, and 
though they are the foundation of all the phenomena of sight, yet, on account 
of their absolute simplicity, they are incapable of analysis, and can never become 
in themselves objects of thought. If we attempt to discover them, we must 



120 THE THEORY OF COLOURS IN RELATION TO COLOUR-BLINDNESS. 

do SO by artificial means ; and our reasonings on tKem must be guided by some 
theory. 

The most general form in which the existing theory can be stated is this, — 
There are certain sensations, finite in number, but infinitely variable in 

degree, which may be excited by the difierent kinds of light. The compound 

sensation resulting from all these is the object of consciousness, is a simple act 

of vision. 

It is easy to see that the numher of these sensations corresponds to what 

may be called in mathematical language the number of independent variables, of 

which sensible colour is a function. 

This will be readily understood by attending to the following cases : — 

1. When objects are illuminated by homogeneous yellow light, the only 
thing which can be distinguished by the eye is difference of intensity or 
brightness. 

If we take a horizontal line, and colour it black at one end, with increasing 
degrees of intensity of yellow light towards the other, then every visible object 
wiU have a brightness corresponding to some point in this line. 

In this case there is nothing to prove the existence of more than one 
sensation in vision. 

In those photographic pictures in which there is only one tint of which 
the different intensities correspond to the different degrees of illumination of the 
object, we have another illustration of an optical effect depending on one variable 
only. 

2. Now, suppose that different kinds of light are emanating from different 
sources, but that each of these sources gives out perfectly homogeneous light, 
then there will be two things on which the nature of each ray will depend : — 
(1) its intensity or brightness ; (2) its hue, which may be estimated by its 
position in the spectrum, and measured by its wave length. 

If we take a rectangular plane, and illuminate it with the different kinds 
of homogeneous light, the intensity at any point being proportional to its hori- 
zontal distance along the plane, and its wave length being proportional to its 
height above the foot of the plane, then the plane will display every possible 
variety of homogeneous light, and will furnish an instance of an optical effect 
depending on two variables. 



THE THEORY OF COLOURS IN RELATION TO COLOUR-BLINDNESS. 



121 



3. Now, let us take the case of nature. We find that colours differ not 
only in intensity and Ime, but also in tint ; that is, they are more or less pure. 
We might arrange the varieties of each colour along a line, which should begin 
with the homogeneous colour as seen in the spectrum, and pass through all 
gradations of tint, so as to become continually purer, and terminate in white. 

We have, therefore, three elements in our sensation of colour, each of which 
may vary independently. For distinctness sake I have spoken of intensity, hue, 
and tint ; but if any other three independent qualities had been chosen, the 
one set might have been expressed in terms of the other, and the results identified. 

The theory which I adopt assumes the existence of three elementary sen- 
sations, by the combination of which all the actual sensations of colour are 
produced. It will be shewn that it is not necessary to specify any given colours 
as typical of these sensations. Young has called them red, green, and violet ; but 
any other three colours might have been chosen, provided that white resulted 
from their combination in proper proportions. 

Before going farther I would observe, that the important part of the theoiy 
is not that three elements enter into our sensation of colour, but that there are 
only three. Optically, there are as many elements in the composition of a ray 
of light as there are different kinds of light in its spectrum; and, therefore, 
strictly speaking, its nature depends on an infinite number of independent 
variables. 

I now go on to the geometrical form into which the theory may be thrown. 
Let it be granted that the three pure sensations corre- 
spond to the colours red, green, and violet, and that we 
can estimate the intensity of each of these sensations 
numerically. 

Let V, r, g be the angular points of a triangle, and 
conceive the three sensations as having their positions at 
these points. If we find the numerical measure of the 
red, green, and violet parts of the sensation of a given 
colour, and then place weights proportional to these parts 

at r, g, and v, and find the centre of gravity of the three weights by the 
ordinary process, that point will be the position of the given colour, and the 
numerical measure of its intensity will be the sum of the tliree primitive 
sensations. 

In this way, every possible colour may have its position and intensity 

VOL. I. 16 




122 THE THEORY OF COLOURS IN RELATION TO COLOUR-BLINDNESS. 

ascertained; and it is easy to see that when two compound colours are com- 
bined, their centre of gravity is the position of the new colour. 

The idea of this geometrical method of investigating colours is to be found 
in Newton's Opticks (Book I., Part 2, Prop. 6), but I am not aware that it has 
been ever employed in practice, except in the reduction of the experiments 
which I have just made. The accuracy of the method depends entirely on the 
truth of the theory of three sensations, and therefore its success is a testimony 
in favour of that theory. 

Every possible colour must be included within the triangle rgv. White 
will be foimd at some point, w, within the triangle. If lines be drawn through 
w to any point, the colour at that point will vary in hue according to the 
angular position of the line drawn to w, and the purity of the tint will depend 
on the length of that line. 

Though the homogeneous rays of the prismatic spectrum are absolutely pure 
in themselves, yet they do not give rise to the "pure sensations" of which we 
are speaking. Every ray of the spectrum gives rise to all three sensations, 
though in different proportions ; hence the position of the colours of the spectrum 
is not at the boundary of the triangle, but in some curve C R Y G B V 
considerably within the triangle. The nature of this curve is not yet determined, 
but may form the subject of a future investigation *. 

All natural colours must be within this curve, and all ordinary pigments 
do in fact lie very much within it. The experiments on the colours of the 
spectrum which I have made are not brought to the same degree of accuracy as 
those on coloured papers. I therefore proceed at once to describe the mode of 
making those experiments which I have found most simple and convenient. 

The coloured paper is cut into the form of discs, each with a small hole 

in the centre, and divided along a radius, so as to admit ^ ^ 

of several of them being placed on the same axis, so that C^^ J 

part of each is exposed. By slipping one disc over another, 

we can expose any given portion of each colour. These >^ — ~^ 
j:«^« „i J „ ^:^.^.^^ j. j.^^i.^4. ,'4.; ^v ( <=> ) 




discs are placed on a little top or teetotum, consisting of \^ y 

a flat disc of tin-plate and a vertical axis of ivory. This 

axis passes through the centre of the discs, and the quantity of each colour exposed 

is measured by a graduation on the rim of the disc, which is divided into 100 parts. 

* [See the author's Memoir in the Philosophical Transactions, 1860, on the Theory o£ Compound 
Colours, and on the relations of the Colours of the Spectrum.] 



THE THEORY OF COLOURS IN RELATION TO COLOUR-BLINDNESS. 123 

By spinning the top, each colour is presented to the eye for a time pro- 
portional to the angle of the sector exposed, and I have found by independent 
experiments, that the colour produced by fast spinning is identical with that 
produced by causing the light of the different colours to fall on the retina at 
once. 

By properly arranging the discs, any given colour may be imitated and 
afterwards registered by the graduation on the rim of the top. The principal 
use of the top is to obtain colour-equations. These are got by producing, by 
two different combinations of colours, the same mixed tint. For this purpose 
there is another set of discs, half the diameter of the others, which lie above 
them, and by which the second combination of colours is formed. 

The two combinations being close together, may be accurately compared, and 
when they are made sensibly identical, the proportions of the different colours 
in each is registered, and the results equated. 

These equations in the case of ordinary vision, are always between four 
colours, not including black. 

From them, by a very simple rule, the different colours and compounds have 
their places assigned on the triangle of colours. The rule for finding the position 
is this : — Assume any three points as the positions of your three standard colours, 
whatever they are ; then form an equation between the three standard colours, 
the given colour and black, by arranging these colours on the inner and outer 
circles so as to produce an identity when spun. Bring the given colour to the 
left-hand side of the equation, and the three standard colours to the right hand, 
leaving out black, then the position of the given colour is the centre of gravity 
of three masses, whose weights are as the number of degrees of each of the 
standard colours, taken positive or negative, as the case may be. 

In this way the triangle of colours may be constructed by scale and compass 
from experiments on ordinary vision. I now proceed to state the results of 
experiments on Colour-Blind vision. 

If we find two combinations of colours which appear identical to a Colour- 
Blind person, and mark their positions on the triangle of colours, then the 
straight line passing through these points will pass through all points corre- 
sponding to other colours, which, to such a person, appear identical-with the first 
two. 

We may in the same way find other lines passing through the series of 

IG— 2 




124 THE THEORY OF COLOURS IN RELATION TO COLOUR-BLINDNESS. 

colours wMch appear alike to the Colour-Blind. All these 
lines either pass through one point or are parallel, ac- 
cording to the standard colours which we have assumed, 
and the other arbitrary assumptions we may have made. 
Knowing this law of Colour-Blind vision, we may predict 
any number of equations which will be true for eyes 
having this defect. 

The mathematical expression of the difference between 
Colour-BUnd and ordinary vision is, that colour to the 
former is a function of two independent variables, but to an ordinary eye, of 
three ; and that the relation of the two kinds of vision is not arbitrary, but 
indicates the absence of a determinate sensation, depending perhaps upon some 
undiscovered structure or organic arrangement, which forms one-third of the 
apparatus by which we receive sensations of colour. 

Suppose the absent structure to be that which is brought most into play 
when red light falls on our eyes, then to the Colour-Blind red light will be 
visible only so far as it affects the other two sensations, say of blue and 
green. It will, therefore, appear to them much less bright than to us, and will 
excite a sensation not distinguishable from that of a bluish-green light. 

I cannot at present recover the results of all my ^periments ; but I recollect 
that the neutral colours for a Colour-Blind person may be produced by com- 
bining 6 degrees of ultramarine with 94 of vermiUon, or 60 of emerald-green 
with 40 of ultramarine. The first of these, I suppose to represent to our eyes 
the kind of red which belongs to the red sensation. It excites the other two 
sensations, and is, therefore, visible to the Colour-BHnd, but it appears very 
dark to them and of no definite colour. I therefore suspect that one of the 
three sensations in perfect vision will be found to correspond to a red of the 
same hue, but of much greater purity of tint. Of the nature of the other two, 
I can say nothing definite, except that one must correspond to a blue, and the 
other to a green, verging to yellow. 

I hope that what I have written may help you in any way in your 
experiments. I have' put down many things simply to indicate a way of thinking 
about colours which belongs to this theory of triple sensation. We are indebted 
to Newton for the original design ; to Young for the suggestion of the means 
of working it out; to Prof. Forbes'' for a scientific history of its application 

*Phil. Mag. 1848. 



THE THEORY OF COLOURS IN RELATION TO COLOUR-BLINDNESS. 125 

to practice; to Helmholtz for a rigorous examination of the facts on which it 
rests; and to Prof Graasman (in the Phil. Mag, for 1852), for an admirable 
theoretical exposition of the subject. The colours given in Hay's Nomenclature 
of Colours are illustrations of a similar theory applied to mixtures of pigments, 
but the results are often different from those in which the colours are combined 
by the eye alone. I hope soon to have results with pigments compared with 
those given by the prismatic spectrum, and then, perhaps, some more definite 
results may be obtained. Yours truly, 

J. C. MAXWELL. 



Edinburgh, 4tli Jan. 1855. 



[From the Transactions of the Royal Society of Edinburgh, Vol xxi. Part ii.] 



VII. Experiments on Colour, as perceived hy the Eye, with remarks on Colour- 
Blindness. Communicated by Dr Gregory. 

The object of tbe following communication is to describe a method by 
which every variety of visible colour may be exhibited to the eye m such a 
form as to admit of accurate comparison ; to shew how experiments so made 
may be registered numerically; and to deduce from these numerical results 
certain laws of vision. 

The different tints are produced by means of a combination of discs of paper, 
painted with the pigments commonly used in the arts, and arranged round an 
axis, so that a sector of any required angular magnitude of each colour may be 
exposed. "When this system of discs is set in rapid rotation, the sectors of 
the different colours become indistinguishable, and the whole appears of one uni- 
form tint. The resultant tints of two different combinations of colours may be 
compared by using a second set of discs of a smaller si^e, and placing these over 
the centre of the first set, so as to leave the outer portion of the larger discs 
exposed. The resultant tint of the first combination will then appear in a ring 
round that of the second, and may be very carefully compared with it. 

The form in which the experiment is most manageable is that of the com- 
mon top. An axis, of which the lower extremity is conical, carries a circular 
plate, which serves as a support for the discs of coloured paper. The circumfer- 
ence of this plate is divided into 100 equal parts, for the purpose of ascertainmg 
the proportions of the different colours which form the combination. When the 
discs have been properly arranged, the upper part of the axis is screwed down, 
so as to prevent any alteration in the proportions of the colours. 

The instrument used in the first series of experiments (at Cambridge, in 
November, 1854) was constructed by myself, with coloured papers procured from 



EXPERIMENTS ON COLOUR, AS PERCEIVED BY THE EYE. 



127 



Mr D. R Hay. The experiments made in the present year were with the 
improved top made by Mr J. M. Bryson, Edinburgh, and coloured papers pre- 
pared by Mr T. Purdie, with the unmixed pigments used in the arts. A number 
of Mr Bryson's tops, with Mr Purdie's coloured papers has been prepared, so as 
to afford different observers the means of testing and comparing results inde- 
pendently obtained. 

The colour used for Mr Purdie's papers were — 



Vermilion 


V 


Ultramarine 


U 


Emerald Green 


EG 


Carmine . 


C 


Prussian Blue . 


PB 


Brunswick Green 


BG 


Red Lead 


RL 


Verditer Blue . 


VB 


Mixture of Ultramarine 




Orange Orpiment 


00 






and Chrome 


uc 


Orange Chrome 


OC 










Chrome Yellow 


CY 










Gamboge 


Gam 










Pale Chrome . 


PC 


Ivory Black . 
Snow White . 


Bk 
SW 







White Paper (Pirie, Aberdeen), 

The colours in the first column are reds, oranges, and yellows; those in 
the second, blues ; and those in the third, greens. Vermilion, ultramarine, and 
emerald green, seem the best colours to adopt in referring the rest to a uniform 
standard. They are therefore put at the head of the Hst, as types of three 
convenient divisions of colour, red, blue, and green. 

It may be asked, why some variety of yellow was not chosen in place of 
green, which is commonly placed among the secondary colours, while yellow 
ranks as a primary? The reason for this deviation from the received system is, 
that the colours on the discs do not represent primary colours at all, but are 
simply specimens of different kinds of paint, and the choice of these was deter- 
mined solely by the power of forming the requisite variety of combinations. Now, 
if red, blue, and yellow, had been adopted, there would have been a difficulty 
in forming green by any compound of blue and yellow, while the yellow formed 
by vermilion and emerald green is tolerably distinct. This will be more clearly 
perceived after the experiments have been discussed, by referring to the diagram. 

As an example of the method of experimenting, let us endeavour to form a 
neutral gray by the combination of vermilion, ultramarine, and emerald green. 
The most perfect results are obtained by two persons acting in concert., when 



128 EXPERIMENTS ON COLOUR, AS PERCEIVED BY THE EYE. 

the operator arranges the colours and spins the top, leaving the eye of the 
observer free from the distracting effect of the bright colours of the papers when 
at rest. 

After placing discs of these three colours on the circular plate of the top, 
and smaller discs of white and black above them, the operator must spin the 
top, and demand the opinion of the observer respecting the relation of the 
outer ring to the inner circle. He will be told that the outer circle is too 
red, too blue, or too green, as the case may be, and that the inner one is too 
light or too dark, as compared with the outer. The arrangement must then be 
changed, so as to render the resultant tint of the outer and inner circles more 
nearly alike. Sometimes the observer will see the inner circle tinted with the 
complementary colour of the outer one. In this case the operator must interpret 
the observation with respect to the outer circle, as the inner circle contains only 
black and white. 

By a little experience the operator will learn how to put his questions, and 
how to interpret their answers. The observer should not look at the coloured 
papers, nor be told the proportions of the colours during the experiments. 
When these adjustments have been properly made, the resultant tints of the 
outer and inner circles ought to be perfectly indistinguishable, when the top 
has a sufficient velocity of rotation. The number of divisions occupied by the 
different colours must then be read off on the edge of the plate, and registered 
in the form of an equation. Thus, in the preceding experiment we have ver- 
milion, ultramarine, and emerald green outside, and black and white inside. The 
numbers, as given by an experiment on the 6th March 1855, in dayhght without 
sun, are — 

•37 V + -27 U + '36 EG = -28 SW+-72 Bk (1). 

The method of treating these equations will be given when we come to the 
theoretical view of the subject. 

In this way we have formed a neutral gray by the combination of the 
three standard colours. We may also form neutral grays of different intensities 
by the combination of vermilion and ultramarine with the other greens, and thus 
obtain the quantities of each necessary to neutralize a given quantity of the 
proposed green. By substituting for each standard colour in succession one of the 
colours which stand under it, we may obtain equations, each of which contains 
two standard colours, and one of the remaining colours. 



EXPERIMENTS ON COLOUR, AS PERCEIVED BY THE EYE. 129 

Thus, in the case of pale chrome, we have, from the same set of experiments, 
•34 PC + -55U + -12 EG = '37 SW + -63Bk (2). 

"We may also make experiments in which the resultiag tint is not a neutral 
gray, but a decided colour. Thus we may combine ultramarine, pale chrome, and 
black, so as to produce a tint identical with that of a compound of vermilion 
and emerald-green. Experiments of this sort are more difficult, both from the 
inability of the observer to express the difference which he detects in two tints 
which have, perhaps, the same hue and intensity, but differ in purity ; and also 
from the complementary colours which are produced in the eye after gazing too 
long at the colours to be compared. 

The best method of arriving at a result in the case before us, is to render 
the hue of the red and green combination something like that of the yellow, to 
reduce the purity of the yellow by the admixture of blue, and to diminish its 
intensity by the addition of black. These operations must be repeated and 
adjusted, till the two tints are not merely varieties of the same colour, but 
absolutely the same. An experiment made 5th March gives — 

•39 PC-I--21 U + -40 Bk = ^59 V-f41 EG (3). 

That these experiments are really evidence relating to the constitution of the 
eye, and not mere comparisons of two things which are in themselves identical, 
may be shewn by observing these resultant tints through coloured glasses, or by 
using gas-light instead of day-light. The tints which before appeared identical 
will now be manifestly different, and will require alteration, to reduce them to 
equality. 

Thus, in the case of carmine, we have by day-light, 

•44 C-h-22 JJ + 'U EG= •I? SW-f-^83 Bk, 
while by gas-light (Edinburgh) 

•47 C-l-^08 U-1-^45 EG = ^25 SW-|-^75 Bk, 
which shews that the yellowing effect of the gas-light teUs more on the white 
than on the combination of colours. If we examine the two resulting tints 
which appeared identical in experiment (3), observing the whirling discs througli 
a blue glass, the combination of yellow, blue, and black, appears redder than- the 
other, while through a yellow glass, the red and green mixture appears redder. 
So also a red glass makes the first side of the equation too dark, and a green 
glass makes it too light. 

VOL. I. 17 



130 EXPERIMENTS ON COLOUR, AS PERCEIVED BY THE EYE. 

The apparent identity of the tints in these experiments is therefore not real, 
but a consequence of a determinate constitution of the eye, and hence arises 
the importance of the results, as indicating the laws of human vision. 

The first result which is worthy of notice is, that the equations, as observed 
by different persons of ordinary vision, agree in a remarkable manner. If care 
be taken to secure the same kind of light in all the experiments, the equations, 
as determined by two independent observers, will seldom shew a difference of 
more than three divisions in any part of the equation containing the bright 
standard colours. As the duller colours are less active in changing the resultant 
tint, their true proportions cannot be so well ascertained. The accuracy of vision 
of each observer may be tested by repeating the same experiment at different 
times, and comparing the equations so found. 

Experiments of this kind, made at Cambridge in November 1854, shew that 
of ten observers, the best were accurate to within 1^ division, and agreed 
within 1 division of the mean of all ; and the worst contradicted themselves to 
the extent of 6 degrees, but still were never more than 4 or 5 from the mean 
of all the observations. 

We are thus led to conclude — 

1st. That the human eye is capable of estimating the likeness of colours 
with a precision which in some cases is very great. 

2nd. That the judgment thus formed is determined, not by the real identity 
of the colours, but by a cause residing in the eye of the observer. 

3rd. That the eyes of different observers vary in accuracy, but agree with 
each other so nearly as to leave no doubt that the law of colour-vision is 
identical for all ordinary eyes. 



Investigation of the Law of the Perception of Colour. 

Before proceeding to the deduction of the elementary laws of the perception 
of colour from the numerical results previously obtained, it will be desirable 
to point out some general features of the experiments which indicate the form 
which these laws must assume. 

Betuming to experiment (1), in which a neutral gray was produced from 
red, blue, and green, we may observe, that, while the adjustments were incom- 



EXPERIMENTS ON COLOUR, AS PERCEIVED BY THE EYE. 131 

plete, the difference of the tints could be detected only by one circle appearing 
more red, more green, or more blue than the other, or by being lighter or 
darker, that is, having an excess or defect of all the three colours together. 
Hence it appears that the nature of a colour may be considered as dependent 
on three things, as, for instance, redness, blueness, and greenness. This is con- 
firmed by the fact that any tint may be imitated by mixing red, blue, and 
green alone, provided that tint does not exceed a certain brilliancy. 

Another way of shewing that colour depends on three things is by con- 
sidering how two tints, say two lilacs, may differ. In the first place, one may 
be lighter or darker than the other, that is, the tints may differ in shade. 
Secondly, one may be more blue or more red than the other, that is, they may 
differ in hue. Thirdly, one may be more or less decided in its colour ; it may vary 
fi*om purity on the one hand, to neutrality on the other. This is sometimes 
expressed by saying that they may differ in tint. 

Thus, in shade, hue, and tint, wo have another mode of reducing the 
elements of colour to three. It will be shewn that these two methods of con- 
sidering colour may be deduced one from the other, and are capable of exact 
numerical comparison. 



On a Geographical Method of Exhibiting the Relations of Colours. 

The method which exhibits to the eye most clearly the results of this theory 
of the three elements of colour, is that which supposes each colour to be repre- 
sented by a point in space, whose distances from three co-ordinate planes are 
proportional to the three elements of colour. But as any method by which the 
operations are confined to a plane is preferable to one recLuiring space of three 
dimensions, we shall only consider for the present that which has been adopted 
for convenience, founded on Newton's Circle of colours and Mayer and Young's 
Triangle. 

Vermilion, ultramarine, and emerald-green, being taken (for convenience) as 
standard colours, are conceived to be represented by three points, taken (for con- 
venience) at the angles of an equilateral triangle. Any colour compounded of 
these three is to be represented by a point found by conceiving masses propor- 
tional to the several components of the colour placed at their respective angular 
points, and taking the centre of gravity of the three masses. In this way, each 

17—2 



132 EXPERIMENTS ON COLOUR, AS PERCEIVED BY THE EYE. 

colour will indicate by its position the proportions of the elements of which it is 
composed. The total intensity of the colour is to be measured by the whole 
number of divisions of V, U, and EG, of which it is composed. This may be 
indicated by a number or coefficient appended to the name of the colour, by 
which the number of divisions it occupies must be multiplied to obtain its mass 
in calculating the results of new combinations. 

This will be best explained by an example on the diagram (No. 1). We 
have, by experiment (l), 

•37 Y+-27 U + -36 EG= -28 SW4- 72 Bk. 

To find the position of the resultant neutral tint, we must conceive a mass 
of -37 at V, of -27 at U, and of '36 at EG, and find the centre of gravity. 
This may be done by taking the line UV, and dividing it in the proportion of 
•37 to ^27 at the point a, where 

aV : aU :: ^27 : '37. 

Then, joining a with EG, divide the joining line in W in the proportion of ^36 
to ("37 + "27), W will be the position of the neutral tint required, which is not 
white, but 0*28 of white, diluted with 0^72 of black, which has hardly any effect 
whatever, except in decreasing the amount of the other colour. The total in- 
tensity of our white paper will be represented by oi = 3'57; so that, whenever 
white enters into an equation, the number of divisions must be multiplied by 
the coefficient 3-57 before any true results can be obtained. 

We may take, as the next example, the method of representing the relation 
of pale chrome to the standard colours on our diagram, by making use of ex- 
periment (2), in which pale chrome, ultramarine, and emerald-green, produced a 
neutral gray. The resulting equation was 

•33PC + -55U + -12EG = -37SW + -63Bk (2). 

In order to obtain the total intensity of white, we must multiply the 
number of divisions, -37, by the proper coefficient, which is 3*57. The result is 
1-32, which therefore measures the total intensity on both sides of the equation. 

Subtracting the intensity of •55U + -12EG, or '67 from 1-32, we obtain '65 
as the corrected value of -33 PC. It will be convenient to use these corrected 
values of the different colours, taking care to distinguish them by small initials 
instead of capitals. 



EXPERIMENTS ON COLOUR, AS PERCEIVED BY THE EYE. 133 

Equation (2) then becomes 

•65 pc + -55 U + -12 EG = 1 -32 w. 

Hence pc must be situated at a point such that w is the centre of gravity 
of •65pc + -55U + '12EG. 

To find it, we begin by determining ^ the centre of gravity of -55 U + '12EG, 
then, joining /8w, the point we are seeking must lie at a certain distance on 
the other side of w from c This distance may be found from the proportion, 

•65 : (-55 + -12) :: ^ : w pc, 
which determines the position of pc. The proper coefficient, by which the ob- 
served vakies of PC must be corrected, is ^, or 1-97. 

We have thus determined the position and coefficient of a colour by a single 
experiment, in which it was made to produce a neutral tint along with two of 
the standard colours. As this may be done with every possible colour, the 
method is applicable wherever we can obtain a disc of the proposed colour. In 
this way the diagram (No. l) has been laid down from observations made in 
daylight, by a good eye of the ordinary type. 

It has been observed that experiments, in which the resultant tint is neutral, 
are more accurate than those in which the resulting tint has a decided colour, 
as in experiment (3), owing to the effects of accidental colours produced in the 
eye in the latter case. These experiments, however, may be repeated till a 
very good mean result has been obtained. 

But since the elements of every colour have been already fixed by our 
previous observations and calculations, the agreement of these results with those 
calculated from the diagram forms a test of the correctness of our method. 

By experiment (No. 3), made at the same time with (l) and (2), we have 
•39PC + -2lU + -40Bk = -59V + -4lEG (3). 

Now, joining XJ with pc, and V with EG, the only common point is that 
at which they cross, namely y. 

Measuring the parts of the line V EG, we find them in the proportion of 
•58 V and "42 EG = 1*00 7. 

Similarly, the line U pc is divided in the proportion 
78 pc and •22U=r00y. 



134 



EXPERIMENTS ON COLOUR, AS PERCEIVED BY THE EYE. 



But -78 pc must be divided by 1-97, to reduce it to PC, as was previously 
explained. The result of calculation is, therefore, 

•39 PC + -22 U + -39 Bk = -58 V + "42 EG, 
the black being introduced simply to fill up the circle. 

This result differs very little from that of experiment (3), and it must be 
recollected that these are single experiments, made independently of theory, and 
chosen at random. 

Experiments made at Cambridge, with all the combinations of five colours, 
shew that theory agrees with calculation always within 0-012 of the whole, 
and sometimes within 0*002. By the repetition of these experiments at the 
numerous opportunities which present themselves, the accuracy of the results 
may be rendered still greater. As it is, I am not aware that the judgments 
of the human eye with respect to colour have been supposed capable of so 
severe a test. 



Further consideration of the Diagram of Colours. 

We have seen how the composition of any tint, in terms of our three 
standard colours, determines its position on the diagram and its proper coefficient. 
In the same way, the result of mixing any other colours, situated at other 
points of the diagram, is to be found by taking the centre of gravity of their 
reduced masses, as was done in the last calculation (experiment 3). 

We have now to turn our attention to the general aspect of the diagram. 

The standard colours, V, U, and EG, occupy the angles of an equilateral 
triangle, and the rest are arranged in the order in which they participate in 
red, blue, and green, the neutral tint being at the point w within the triangle. 
If we now draw lines through w to the different colours ranged round it, we 
shall find that, if we pass from one line to another in the order in which they 
lie from red to green, and through blue back again to red, the order will be — 



Carmine . 
Vermilion . 
Red Lead . 
Oi-ange Orpiment 
Orange Chrome 
Chrome Yellow 
Gramboge . 



Coefficient. 




0-4 


Pale Chrome 


1-0 


Mixed Green (U C) 


1-3 


Brunswick Green 


10 


Emerald Green . 


1-6 


Verditer Blue . 


1-5 


Prussian Blue . 


1-8 


Ultramarine 



Coefficient. 
2 
0-4 
0-2 
10 
0-8 
01 
10 



EXPERIMENTS ON COLOUR, AS PERCEITED BY THE EYE. 135 

It may be easily seen that this arrangement of the colours corresponds to 
that of the prismatic spectrum ; the only difference being that the spectrum 
is deficient in those fine purples which lie between ultramarine and vermilion, 
and which are easily produced by mixture. The experiments necessary for deter- 
mining the exact relation of this list to the lines in the spectrum are not yet 
completed. 

If we examine the colours represented by different points in one of these 
lines through w, we shall find the purest and most decided colours at its outer 
extremity, and the faint tints approaching to neutrality nearer to w. 

If we also study the coefficients attached to each colour, we shall find that 
the brighter and more luminous colours have higher numbers for their coefficients 
than those which are dark. 

In this way, the qualities which we have already distinguished as hue, tint, 
and shade, are represented on the diagram by angular position with respect to ir, 
distance from w, and coefficient; and the relation between the two methods of 
reducing the elements of colour to three becomes a matter of geometry. 



Theory of the Perception of Colour. 

Opticians have long been divided on this point ; those who trusted to 
popular notions and their own impressions adopting some theory of three primary 
colours, while those who studied the phenomena of light itself proved that no 
such theory could explain the constitution of the spectrum. Newton, who was 
the first to demonstrate the actual existence of a series of kinds of light, 
countless in number, yet all perfectly distinct, was also the first to propound 
a method of calculating the effect of the mixture of various coloured light ; 
and this method was substantially the same as that which we have just 
verified. It is true, that the directions which he gives for the construction 
of his circle of colours are somewhat arbitrary, being probably only intended 
as an indication of the general nature of the method, but the method itself 
is mathematically reducible to the theory of three elements of the colour- 
sensation*. 

♦ See Note III. For a confirmation of Newton's analysis of Light, see Helmholtz, Pogg. Ann, 
1852; and Phil. Mag. 1852, Part ii. 



136 EXPERIMENTS ON COLOUR, AS PERCEIVED BY THE EYE. 

Youno", who made the next great step in the establishment of the theory 
of light, seems also to have been the first to follow out the necessary conse- 
quences of Newton's suggestion on the mixture of colours. He saw that, since 
this tripUcity has no foundation in the theory of light, its cause must be looked 
for in the constitution of the eye; and, by one of those bold assumptions 
which sometimes express the result of speculation better than any cautious 
trains of reasoning, he attributed it to the existence of three distinct modes 
of sensation in the retina, each of which he supposed to be produced in different 
deo-rees by the different rays. These three elementary effects, according to his 
view, correspond to the three sensations of red, green, and violet, and would 
separately convey to the sensorium the sensation of a red, a green, and a violet 
picture ; so that by the superposition of these pictures, the actual variegated 
world is represented*. 

In order fully to understand Young's theory, the function which he 
attributes to each system of nerves must be carefully borne in mind. Each nerve 
acts, not, as some have thought, by conveying to the mind the knowledge of the 
length of an undulation of light, or of its periodic time, but simply by being 
Quore or less affected by the rays which fall on it. The sensation of each 
elementary nerve is capable only of increase and diminution, and of no other 
change. We must also observe, that the nerves corresponding to the red 
sensation are affected chiefly by the red rays, but in some degree also by those 
of every other part of the spectrum ; just as red glass transmits red rays freely, 
but also suffers those of other colours to pass in smaller quantity. 

This theory of colour may be illustrated by a supposed case taken from 
the art of photography. Let it be required to ascertain the colours of a land- 
scape, by means of impressions taken on a preparation equally sensitive to rays of 
every colour. 

Let a plate of red glass be placed before the camera, and an impression 
taken. The positive of this will be transparent wherever the red light has been 
abundant in the landscape, and opaque where it has been wanting. Let it now 
be put in a magic lantern, along with the red glass, and a red picture will be 
thrown on the screen. 

Let this operation be repeated with a green and a violet glass, and, by 

* Young's Lectures, p. 345, Kelland's Edition. See also Helmholtz's statement of Young's Theory, 
in his Paper referred to in Note I. ; and Herschel's LigJU, Art. 518. 



EXPERIMENTS ON COLOUR, AS PERCEIVED BY THE EYE. 137 

means of three magic lanterns, let the three images be superimposed on the 
screen. The colour of any point on the screen will then depend on that of the 
corresponding point of the landscape; and, by properly adjusting the intensities 
of the lights, &c., a complete copy of the landscape, as far as visible colour is 
concerned, will be thrown on the screen. The only apparent difference will be, 
that the copy will be more subdued, or less pure in tint, than the original. 
Here, however, we have the process performed twice — first on the screen, and 
then on the retina. 

This illustration will shew how the functions which Young attributes to the 
three systems of nerves may be imitated by optical apparatus. It is therefore 
unnecessary to search for any direct connection between the lengths of the 
undulations of the various rays of light and the sensations as felt by us, as 
the threefold partition of the properties of light may be effected by physical 
means. The remarkable correspondence between the results of experiments on 
different individuals would indicate some anatomical contrivance identical in all. 
As there is little hope of detecting it by dissection, we may be content at 
present with any subsidary evidence which we may possess. Such evidence is 
furnished by those individuals who have the defect of vision which was 
described by Dalton, and which is a variety of that which Dr G. Wilson has 
lately investigated, under the name of Colour-Blindness. 



Testimony of the Colour- Blind with respect to Colour. 

Dr George Wilson has described a great number of cases of colour- 
bhndness, some of which involve a general indistinctness in the appreciation 
of colour, while in others, the errors of judgment are plainly more numerous 
in those colours which approach to red and green, than among those which 
approach to blue and yellow. In these more definite cases of colour-blindness, 
the phenomena can be tolerably well accoimted for by the hypothesis of an 
insensibility to red light; and this is, to a certain extent, confirmed by the 
fact, that red objects appear to these eyes decidedly more obscure than to 
ordinary eyes. But by experiments made with the pure spectrum, it appears 
that though the red appears much more obscure than other colours, it is not 
wholly invisible, and, what is more curious, resembles the green more than 
any other colour. The spectrum to them appears faintly luminous in the red; 

VOL. L 18 



138 EXPERIMENTS ON COLOUR, AS PERCEIVED BY THE EYE. 

bright yellow from orange to yellow, bright but not coloured from yellow- 
green to blue, and then strongly coloured in the extreme blue and violet, 
after which it seems to approach the neutral obscure tint of the red. It is 
not easy to see why an insensibihty to red rays should deprive the green 
rays, which have no optical connection with them, of their distinctive appearance. 
The phenomena seem rather to lead to the conclusion that it is the red 
serisation which is wanting, that is, that supposed system of nerves which is 
affected in various degrees by all light, but chiefly by red. We have fortunately 
the means of testing this hypothesis by numerical results. 

Of the subjects of my experiments at Cambridge, four were decided cases 
of colour-bHndness. Of these two, namely, Mr E. and Mr S., were not 
suflficiently critical in their observations to afford any results consistent within 
10 divisions of the colour-top. The remaining two, Mr N. and Mr X., were 
as consistent in their observations as any persons of ordinary vision can be, 
while the results shewed all the more clearly how completely their sensations 
must differ from ours. 

The method of experimenting was the same as that adopted with ordinary 
eyes, except that in these cases the operator can hardly influence the result 
by yielding to his own impressions, as he has no perception whatever of the 
similarity of the two tints as seen by the observer. The questions which he 
must ask are two, Which circle appears most blue or yellow ? Which appears 
lightest and which darkest ? By means of the answers to these questions he 
must adjust the resulting tints to equality in these respects as it appears to 
the observer, and then ascertain that these tints now present no difference of 
colour whatever to his eyes. The equations thus obtained do not require five 
colours including black, but four only. For instance, the mean of several obser- 
vations gives — 

•19 G+'05 B + -76 Bk=100R (4). 

[In these experiments R, B, G, Y, stand for red, blue, green, and yellow 
papers prepared by Mr D. R. Hay. I am not certain that they are identical 
with his standard colours, but I beUeve so. Their relation to vermihon, ultra- 
marine, and emerald-green is given in diagram (1). Their relations to each other 
are very accurately given in diagram (2).] 

It appears, then, that the dark blue-green of the left side of the equation 
is equivalent to the full red of the right side. 



EXPERIMENTS ON COLOUR, AS PERCEIVED BY THE EYE. 139 

Hence, if we divide the line BG in the proportion 19 to 5 at the point y8, 
and join R)8, the tint at ^ will differ from that at R (to the colour-blind) 
only in being more brilliant in the proportion of 100 to 24, and all inter- 
mediate tints on the line R^ will appear to them of the same hue, but 
of intermediate intensities. 

Now, if we take a point D, so that RD is to R^ in the proportion of 
24 to 100 — 24, or 76, the tint of D, if producible, should be invisible to 
the colour-blind. D, therefore, represents the pure sensation which is unknown 
to the colour-blind, and the addition of this sensation to any others cannot 
alter it in their estimation. It is for them equivalent to black. 

Hence, if we draw lines through D in different directions, the colours 
belonging to any line ought to differ only in intensity as seen by them, so 
that one of them may be reduced to the other by the addition of black 
only. If we draw DW and produce it, all colours on the upper side of DW 
will be varieties of blue, and those on the under side varieties of yellow, so 
that the line DW is a boundary line between their two kinds of colour, blue 
and yellow being the names by which they call them. 

The accuracy of this theory will be evident from the comparison of the 
experiments which I had an opportunity of making on Mr N. and Mr X. with 
each other, and with measurements taken from the diagram No. 2, which was 
constructed from the observations of ordinary eyes only, the point D alone 
being ascertained from a series of observations by Mr N. 

Taking the point y, between R and B, it appears, by measurement of the 
lines Ry and By, that y corresponds to 

•07 B + -93R. 

By measurement of Wy and Dy, and correction by means of the coeflScient 
of W, and caUing D black in the colour-blind language, y corresponds to 

•105 W-f895 Bk. 
Therefore 

By measurement -93 R+ '07 B = ^105 W + •sgs Bk 1 

By observation N. & X. together "94 R-f -06 B = •lO W-f-^90 Bk I (5). 

By X. alone -93 R-h-07 B = -10 W + -90 Bk J 

The agreement here is as near as can be expected. 

18—2 



140 



EXPERIMENTS ON COLOira, AS PERCEIVED BY THE EYE. 



By a similar calculation with respect to the point 8, between B and G, 

By measurement -43 B + -57 G = -335 W + *665 Bk 1 

Observed by N. and X '41 B + '59 G = '34 W + -66 Bk I (6). 

By X. alone -42 B + -58 G = -32 W + -68 Bk J 

We may also observe, that the line GD crosses RY. At the point of inter- 
section we have — 

By calculation '87 B + 'IS Y = -34 G + -66 Bk 

Observed by N. and X -86 R + -14 Y = -40 G + 'GO Bk 

X •84R + '16 Y=-31 G + '69 Bk 

X -QOR + 'IO Y = -27 G + 73Bk 



.(7). 



Here observations are at variance, owing to the decided colours produced 
affecting the state of the retina, but the mean agrees well with calculation. 

Drawing the line BY, we find that it cuts lines through D drawn to every 
colour. Hence all colours appear to the colour-blind as if composed of blue 
and yellow. By measurement on the diagram, we find for red 

Measured -138 Y+-123 B + 749 Bk = 100 R' 

Observed by N..., -15 Y + 'll B-1--74 Bk = 100RJ- (8). 

X....-13 Y + 'll B + -76 Bk = 100R 



.(9). 



For green we have in the same way — 

Measured 705 Y + -295 B = '95 G + -05 Bkl 

Observed by N.... 70 Y + -30 B = -86 G + -14 Bk i .... 
X.... 70 Y+-30 B = '83 G+-17BkJ 

For white — 

Measured '407 Y + -593 B = '326 W + "674 Bk 

Observed by N.... -40 Y+-60 B = -33 W+-67 Bk 
X.... -44 Y+-56 B=-33 W+-67 Bk 

The accuracy of these results shews that, whether the hypothesis of the 
want of one element out of three necessary to perfect vision be actually true 
or not, it affords a most trustworthy foundation on which to build a theory 
of colour-blindness, as it expresses completely the observed facts of the case. 
They also furnish us with a datum for our theory of perfect vision, namely, 
the point D, which points out the exact nature of the colour-sensation, which 
must be added to the colour-blind eye to render it perfect. I am not aware 



EXPERIMENTS ON COLOUR, AS PERCEIVED BY THE EYE. 141 

of any method of determining by a legitimate process the nature of the other 
two sensations, although Young's reasons for adopting something like green and 
violet appear to me worthy of attention. 

The only remaining subject to which I would call the attention of the 
Society is the effect of coloured glasses on the colour-blind. Although they can- 
not distinguish reds and greens from varieties of gray, the transparency of red 
and green glasses for those kinds of light is very different. Hence, after finding 
a case such as that in equation (4), in which a red and a green appear iden- 
tical, on looking through a red glass they see the red clearly and the green 
obscurely, while through a green glass the red appears dark and the green light. 

By furnishing Mr X. with a red and a green glass, which he could dis- 
tinguish only by their shape, I enabled him to make judgments in previously 
doubtful cases of colour with perfect certainty. I have since had a pair of 
spectacles constructed with one eye-glass red and the other greeiL These Mr X. 
intends to use for a length of time, and he hopes to acquire the habit of discri- 
minating red from green tints by their different effects on his two eyes. Though 
he can never acquire our sensation of red, he may then discern for himself what 
things are red, and the mental process may become so familiar to him as to act 
unconsciously like a new sense. 

In one experiment, after looking at a bright light, with a red glass over one 
eye and a green over the other, the two tints in experiment (4) appeared to him 
altered, so that the outer circle was lighter according to one eye, and the inner 
according to the other. As far as I could ascertain, it appeared as if the eye 
which had used the red glass saw the red circle brightest. This result, which 
seems at variance with what might be expected, I have had no opportunity of 
verifying. 

This paper is already longer than was originally intended For further 
information I would refer the reader to Newton's Optich, Book i. Part ii., to 
Young's Lectures on Natural Philosophy, page 345, to Mr D. R. Hay's works on 
Colours, and to Professor Forbes on the "Classification of Colours" (Phil. Mag., 
March, 1849). 

The most remarkable paper on the subject is that of M. Helmholtz, in the 
Philosophical Magazine for 1852, in which he discusses the different theories of 
primary colours, and describes his method of mixing the colours of the spectrum. 
An examination of the results of M. Helmholtz with reference to the theory 



142 EXPERIMENTS OX COLOUR, AS PERCBIV^ED BY THE EYE. 

of three elements of colour, by Professor Grassmann, is translated in the Phil. 
Mag., April, 1854. 

References to authors on colour-blindness are given in Dr G. Wilson's papers 
on that subject. A valuable Letter of Sir J. F. W. Herschel to Dalton on his 
peculiarity of vision, is to be found in the Life of Dalton by Dr Henry. 

I had intended to describe some experiments on the propriety of the method 
of mixino- colours by rotation, which might serve as an extension of Mr Swan's 
experiments on instantaneous impressions on the eye. These, together with the 
explanation of some phenomena which seem to be at variance with the theory of 
vision here adopted, must be deferred for the present. On some future occasion, 
I hope to be able to connect these simple experiments on the colours of pigments 
with others in which the pure hues of the spectrum are used. I have already 
constructed a model of apparatus for this purpose, and the results obtained are 
sufficiently remarkable to encourage perseverance. 



Note I. 
On different Methods of Exhibiting the Mixtures of Colours. 

(1) Mechanical Mixture of Coloured Powders. 
By grinding coloured powders together, the differently- coloured particles may 
be so intermingled that the eye cannot distinguish the colours of the separate 
powders, but receives the impression of a uniform tint, depending on the nature 
and proportions of the pigments used. In this way, Newton mixed the powders 
of orpiment, purple, bise, and viride ceris, so as to form a gray, which, in sun- 
light, resembled white paper in the shade. (Newton's Opticks, Book i. Part n., 
Exp. XV.) This method of mixture, besides being adopted by all painters, has 
been employed by optical writers as a means of obtaining numerical results. 
The specimens of such mixtures given by B. R. Hay in his works on Colour, 
and the experiments of Professor J. D. Forbes on the same subject, shew the 
importance of the method as a means of classifying colours. There are two 
objections, however, to this method of exhibiting colours to the eye. When 
two powders of unequal fineness are mixed, the particles of the finer powder 
cover over those of the coarser, so as to produce more than their due effect 
in influencing the resultant tint. For instance, a small quantity of lamp-black. 



EXPERIMENTS ON COLOUR, AS PERCEIVED BY THE EYE. 143 

mixed with a large quantity of chalk, will produce a mixture which is nearly 
black. Although the powders generally used are not so different in this respect 
as lamp-black and chalk, the results of mixing given weights of any coloured 
powders must be greatly modified by the mode in which these powders have 
been prepared. 

Again, the light which reaches the eye from the surface of the mixed pow- 
ders consists partly of light which has fallen on one of the substances mixed 
without being modified by the other, and partly of light which, by repeated 
reflection or transmission, has been acted on by both substances. The colour of 
these rays will not be a mixture of those of the substances, but will be the 
result of the absorption due to both substances successively. Thus, a mixture of 
yellow and blue produces a neutral tint tending towards red, but the remainder 
of white light, after passing through both, is green; and this green is generally 
sufficiently powerful to overpower the reddish gray due to the separate colours 
of the substances mixed. This curious result has been ably investigated by 
Professor Helmholtz of Konigsberg, in his Memoir on the Theory of Compound 
Colours, a translation of which may be found in the Annals of Philosophy for 
1852, Part 2. 

(2) Mixture of differently-coloured Beams of Light by Superposition 

on an Opaque Screen. 
When we can obtain light of sufficient intensity, this method produces the 
most beautiful results. The best series of experiments of this kind are to be 
found in Newton's Opticks, Book i. Part ii. The different arrangements for 
mixing the rays of the spectrum on a screen, as described by Newton, form 
a very complete system of combinations of lenses and prisms, by which almost 
every possible modification of coloured light may be produced. The principal 
objections to the use of this method are— (1) The difficulty of obtaining a con- 
stant supply of uniformly intense light; (2) The uncertainty of the effect of 
the position of the screen with respect to the incident beams and the eye of 
the observer; (3) The possible change in the colour of the incident light due 
to the fluorescence of the substance of the screen. Professor Stokes haa found 
that many substances, when illuminated by homogeneous light of one refrangi- 
bility, become themselves luminous, so as to emit light of lower refrangibility. 
This phenomenon must be carefully attended to when screens are used to exhibit 
light. 



144 EXPERIMENTS ON COLOUK, AS PERCEIVED BY THE EYE. 



(3) Union of Coloured Beams hy a Piism so as to form one Beam. 

The mode of viewing the beam of light directly, without first throwing it 
on a screen, was not much used by the older experimenters, but it possesses 
the advantage of saving much light, and admits of examining the rays before 
they have been stopped in any way. In Newton's 11th proposition of the 2nd 
Book, an experiment is described, in which a beam is analysed by a prism, 
concentrated by a lens, and recombined by another prism, so as to form a beam 
of white light similar to the incident beam. By stopping the coloured rays at 
the lens, any proposed combination may be made to pass into the emergent 
beam, where it may be received directly by the eye, or on a screen, at pleasure. 

The experiments of Helmholtz on the colours of the spectrum were made 
with the ordinary apparatus for directly viewing the pure spectrum, two oblique 
slits crossing one another being employed to admit the light instead of one 
vertical sht. Two pure spectra were then seen crossing each other, and so 
exhibiting at once a large number of combinations. The proportions of these 
combinations were altered by varying the inclination of the slits to the plane of 
lefraction, and in this way a number of very remarkable results were obtained, — 
for which see his Memoir, before referred to. 

In experiments of the same kind made by myself in August 1852 (inde- 
pendently of M. Helmholtz), I used a combination of three moveable vertical 
slits to admit the light, instead of two cross shts, and observed the compound 
ray through a slit made in a screen on which the pure spectrum is formed. 
In this way a considerable field of view was filled with the mixed light, and 
might be compared with another part of the field illuminated by light proceeding 
from a second system of slits, placed below the first set. The general character 
of the results agreed with those of M. Helmholtz. The chief difficulties seemed 
to arise from the defects of the optical apparatus of my own eye, which ren- 
dered apparent the compound nature of the light, by analysing it as a prism 
or an ordinary lens would do, whenever the lights mixed differed much in 
refrangibility. 



EXPERIMENTS ON COLOUR, AS PERCEIVED BY THE EYE. 145 



(4) Union of two beams by means of a transparent surface, which reflects 
the first and transmits the second. 

The simplest experiment of this kind is described by M. Helmholtz. He 
places two coloured wafers on a table, and then, taking a piece of transparent 
glass, he places it between them, so that the reflected image of one apparently 
coincides with the other as seen through the glaas. The colours are thus mixed, 
and, by varying the angle of reflection, the relative intensities of the reflected 
and transmitted beams may be varied at pleasure. 

In an instrument constructed by myself for photometrical purposes two re- 
flecting plates were used. They were placed in a square tube, so as to polarize 
the incident light, which entered through holes in the sides of the tubes, and 
was reflected in the direction of the axis. In this way two beams oppositely 
polarized were mixed, either of which could be coloured in any way by coloured 
glasses placed over the holes in the tube. By means of a Nicol's prism placed 
at the end of the tube, the relative intensities of the two colours as they 
entered the eye could be altered at pleasure. 

(5) Union of two coloured beams by means of a doubly -refracting Prism. 

I am not aware that this method has been tried, although the opposite 
polarization of the emergent rays is favourable to the variation of the experiment. 



(6) Successive presentation of the different Colours to the Retina. 

It has long been known, that light does not produce its full effect on the 
eye at once, and that the effect, when produced, remains visible for some time 
after the light has ceased to act. In the case of the rotating disc, the various 
colours become indistinguishable, and the disc appears of a imiform tint, which 
is in some sense the resultant of the colours so blended. This method of com- 
bining colours has been used since the time of Newton, to exhibit the results 
of theory. The experiments of Professor J. D. Forbes, which I witnessed in 
1849, first encouraged me to think that the laws of this kind of mixture might 
be discovered by special experiments. After repeating the well-known experiment 
in which a series of colours representing those of the spectrum are combined 

VOL. I. ^^ 



146 EXPERIMENTS ON COLOUB, AS PERCEIVED BY THE EYE. 

to form gray, Professor Forbes endeavoured to form a neutral tint, by the 
combination of three colours only. For this purpose, he combined the three 
so-called primary colours, red, blue, and yellow, but the resulting tint could 
not be rendered neutral by any combination of these colours ; and the reason 
was found to be, that blue and yellow do not make green, but a pinkish tint, 
when neither prevails in the combination. It was plain, that no addition of 
red to this, could produce a neutral tint. 

This result of mixing blue and yellow was, I beUeve, not previously known. 
It directly contradicted the received theory of colours, and seemed to be at 
variance with the fact, that the same blue and yellow paint, when ground 
together, do make green. Several experiments were proposed by Professor Forbes, 
in order to eliminate the effect of motion, but he was not then able to under- 
take them. One of these consisted in viewing alternate stripes of blue and 
yellow, with a telescope out of focus. I have tried this, and find the resultant 
tint pink as before*. I also found that the beams of light coloured by trans- 
mission through blue and yellow glasses appeared pink, when mixed on a screen, 
while a beam of light, after passing through both glasses, appeared green. By 
the help of the theory of absorption, given by Herschelf, I made out the 
complete explanation of this phenomenon. Those of pigments were, I think, first 
explained by Helmholtz in the manner above referred to J. 

It may still be asked, whether the effect of successive presentation to the 
eye is identical with that of simultaneous presentation, for if there is any action 
of the one kind of light on the other, it can take place only in the case of 
vsimultaneous presentation. An experiment tending to settle this point is recorded 
by Newton (Book i. Part ii., Exp. 10). He used a comb with large teeth to 
intercept various rays of the spectrum. When it was moved slowly, the various 
colours could be perceived, but when the speed was increased the result was 
perfect whiteness. For another form of this experiment, see Newton's Sixth 
Letter to Oldenburg (Horsley's Edition, Vol. iv., page 335). 

In order more fully to satisfy myself on this subject, I took a disc in 
which were cut a number of sUts, so as to divide it into spokes. In a plane, 
net-rly passing through the axis of this disc, I placed a blue glass, so that one 

* See however Encyc. Metropolitana, Art. "Light," section 502. t lb. sect. 516. 

X I have lately seen a passage in Moigno's Cosmos, stating that M. Plateau, in 1819, had obtained 
jjray by whirling together gamboge and Prussian blue. Correspondance Math, et Phys. de M. Quet«let, 
Vol. v., p. 221. 



EXPERIMENTS ON COLOUR, AS PERCEIVED BY THE EYE. 147 

half of the disc might be seen by transmitted light — blue, and the other by 
reflected light — white. In the course of the reflected light I placed a yellow 
glass, and in this way I had two nearly coincident images of the slits, one 
yellow and one blue. By turning the disc slowly, I observed that in some 
parts the yellow slits and the blue slits appeared to pass over the field alter- 
nately, while in others they appeared superimposed, so as to produce alternately 
their mixture, which was pale pink, and complete darkness. As long as the 
disc moved slowly I could perceive this, but when the speed became great, the 
whole field appeared uniformly coloured pink, so that those parts in which the 
colours were seen successively were indistinguishable from those in which they 
were presented together to the eye. 

Another form in which the experiment may be tried requires only the 
colour-top above described. The disc should be covered with alternate sectors 
of any two colours, say red and green, disposed alternately in four quadrants. 
By placing a piece of glass above the top, in the plane of the axis, we make 
the image of one half seen by reflection coincide with that of the other seen 
by transmission. It wiU then be seen that, in the diameters of the top which 
are parallel and perpendicular to the plane of reflection, the transmitted green 
coincides with the reflected green, and the transmitted red with the reflected 
red, so that the result is always either pure red or pure green. But in the 
diameters intermediate to these, the transmitted red coincides with the reflected 
green, and vice versa, so that the pure colours are never seen, but only their 
mixtures. As long as the top is spun slowly, these parts of the disc will 
appear more steady in colour than those in which the greatest alternations 
take place ; but when the speed is sufficiently increased, the disc appears per- 
fectly uniform in colour. From these experiments it appears, that the apparent 
mixture of colours is not due to a mechanical superposition of vibrations, or 
to any mutual action of the mixed rays, but to some cause residing in the 
constitution of the apparatus of vision. 



(7) Presentation of the Colours to he mixed one to each Eye. 

This method is said not to succeed with some people ; but I have always 
found that the mixture of colours was perfect, although it was difficult to con- 
ceive the objects seen by the two eyes as identical. In using the spectacles, 

19—2 



148 EXPERIMENTS ON COLOUR, AS PERCEIVED BY THE EYE. 

of which one eye is green and the other red, I have found, when looking at 
an arrangement of green and red papers, that some looked metallic and others 
transparent. This arises from the very different relations of brightness of the 
two colours as seen by each eye through the spectacles, which suggests the false 
conclusion, that these differences are the result of reflection from a polished 
surface, or of light transmitted through a clear one. 



Note IT. 

Results of Experiments with Mr Hay's Papers at Cambridge, November, 1854. 

The mean of ten observations made by six observers gave 

•449 E+-299 G + -252 B=-224 W+776 Bk (l). 

■696 R+-304 G = '181 B + -327 Y + '492 Bk (2). 

These two equations served to determine the positions of white and yellow 
in diagram No. 2. The coeflScient of W is 4*447, and that of yellow 2'506. 

From these data we may deduce three other equations, either by calcu- 
lation, or by measurement on the diagram (No. 2). 

Eliminating green from the equations, we find 

•565 B + -435 Y = -307 E. + -304 W + -389 Bk (3). 

The mean of three observations by three different observers gives 

•573 B-f477 Y = ^313 E + ^297 W + -390Bk. 
Errors of calculation - '008 B + ^008 Y - '006 K + ^007 W - •OOl Bk. 

The point on the diagram to which this equation corresponds is the intersec- 
tion of the lines BY and RW, and the resultant tint is a pinkish-gray. 

Eliminating red from the equations, we obtain 
Calculation "533 B-fl50 G-f317 Y = ^337 W-f -663 Bk" 

By 10 observations -537 B-l- '146 G-h ^317 Y= -337 W-f '663 Bk ■ (4). 

Errors -'004 -f- -004 — — — 

Eliminating blue •660 R-f340 G = -218 Y + -108 W-f '682 Bkl 

By 5 observations ^672 R-f '328 G = "224 Y+ '094 W-f672 Bk i (5). 

Errors -'012 -f012 -•006 -f014 -f008 I 



EXPERIMENTS ON COLOUR, AS PERCEIVED BY THE EYE. 149 

Note III. 

On the Tlicory of Compound Colours. 

Newton's theorem on the mixture of colours is to be found in his Opticks, 
Book I., Part ii., Prop. vi. 

In a mixtiu'e of primary colours^ the quantity and quality of each being 
gicen, to know the colour of the compound. 

He divides the circumference of a circle into parts proportional to the seven 
musical intervals, in accordance with his opinion of the divisions of the spectrum. 
He then conceives the colours of the spectrum arranged round the circle, and at 
the centre of gravity of each of the seven arcs he places a little circle, the 
area of which represents the number of rays of the corresponding colour which 
enter into the given mixture. He takes the centre of gravity of all these circles 
to represent the colour formed by the mixture. The hue is determined by 
drawing a line through the centre of the circle and this point to the circum- 
ference. The position of this line points out the colour of the spectrum which 
the mixture most resembles, and the distance of the resultant tint from the 
centre determines the fulness of its colour. 

Newton, by this construction (for which he gives no reasons), plainly shews 
that he considered it possible to find a place within his circle for every possible 
colour, and that the entire nature of any compound colour may be known from 
its place in the circle. It will be seen that the same colour may be compounded 
from the colours of the spectrum in an infinite variety of ways. The apparent 
identity of all these mixtures, which are optically different, as may be shewn by 
the prism, implies some law of vision not explicitly stated by Newton. This 
law, if Newton's method be true, must be that which I have endeavoured to 
establish, namely, the threefold nature of sensible colour. 

With respect to Newton's construction, we now know that the proportions 
of the colours of the spectrum vary with the nature of the refracting medium. 
The only absolute index of the kind of light is the time of its vibration. The 
length of its vibration depends on the medium in which it is ; and if any pro- 
portions are to be sought among the wave-lengths of the colours, they must 
be determined for those tissues of the eye in which their physical effects are 



150 EXPERIMENTS ON COLOUR, AS PERCEIVED BY THE EYE. 

supposed to terminate. It may be remarked, *that the apparent colour of the 
spectrum changes most rapidly at three points, which lie respectively in the 
yellow, between blue and green, and between violet and blue. The wave-lengths 
of the corresponding rays in 'water are in the proportions of three geometric 
means between 1 and 2 very nearly. This result, however, is not to be con- 
sidered established, unless confirmed by better observations than mine. 

The only safe method of completing Newton's construction is by an exami- 
nation of the colours of the spectrum and their mixtures, and subsequent 
calculation by the method used in the experiments with coloured papers. In 
this way I hope to determine the relative positions in the colour-diagram of 
every ray of the spectrum, and its relative intensity in the solar light. The 
spectrum will then form a curve not necessarily circular or even re-entrant, and 
its peculiarities so ascertained may form the foundation of a more complete 
theory of the colour-sensation. 



On the relation of the pure rays of the Spectrum to the three assumed Elementary 

Sensations. 

If we place the three elementary colour-sensations (which we may call, after 
Young, red, green, and violet) at the angles of a triangle, all colours which 
the eye can possibly perceive (whether by the action of light, or by pressure, 
disease, or imagination) must be somewhere within this triangle, those which lie 
farthest from the centre being the fullest and purest colours. Hence the colours 
which lie at the middle of the sides are the purest of their kind which the 
eye can see, although not so pure as the elementary sensations. 

It is natural to suppose that the pure red, green, and violet rays of the 
spectrum produce the sensations which bear their names in the highest purity. 
But from this supposition it would follow that the yellow, composed of the red 
and green of the spectrum, would be the most intense yellow possible, while 
it is the result of experiment, that the yellow of the spectrum itself is much 
more full in colour. Hence the sensations produced by the pure red and green 
rays of the spectrum are not the pure sensations of our theory. Newton has 
remarked, that no two colours of the spectrum produce, when mixed, a colour 
equal in fulness to the intermediate colour. The colours of the spectrum are 
all more intense than any compound ones. Purple is the only colour which 



EXrERIMENTS ON COLOUR, AS PERCEIVED BY TUE EYE. 151 

must be produced by combination. The experiments of Helmholtz lead to the 
same conclusion ; and hence it would appear that we can find no part of the 
spectrum which produces a pure sensation. 

An additional, though less satisfactory evidence of this, is supplied by the 
observation of the colours of the spectrum when excessively bright. They then 
appear to lose their peculiar colour, and to merge into pure whiteness. This 
is probably due to the want of capacity of the organ to take in so strong an 
impression ; one sensation becomes first saturated, and the other two speedily 
follow it, the final efiect being simple brightness. 

From these facte I would conclude, that every ray of the spectrum is capable 
of producing all three pure sensations, though in different degrees. The curve, 
therefore, which we have supposed to represent the spectrum will be quite within 
the triangle of colour. All natural or artificial colours, being compounded of 
the colours of the spectrum, must lie within this curve, and, therefore, the colours 
corresponding to those parts of the triangle beyond this curve must be for ever 
unknown to us. The determination of the exact nature of the pure sensations, 
or of their relation to ordinary colours, is therefore impossible, unless we can 
prevent them from interfering with each other as they do. It may be possible 
to experience sensations more pure than those directly produced by the spec- 
trum, by first exhausting the sensibility to one colour by protracted gazing, and 
then suddenly turning to its opposite. But if, as I suspect, colour-blindness be 
due to the absence of one of these sensations, then the point D in diagram (2), 
which indicates their absent sensation, indicates also our pure sensation, which 
we may call red, but which we can never experience, because all kinds of 
light excite the other sensations. 

Newton has stated one objection to his theory, as follows: — "Also, if only 
two of the pnmanj colours, which in tJw circle are opposite to one another, be 
mixed in an equal proportion, the point Z" (the resultant tint) "shall fall upon 
the centre " (neutral tint) ; " and yet the colour compounded of these two shcdl 
not he p>erfectly white, hut some faint anonymous colour. For I could never yet, by 
mixing only two primary colours, produce a perfect ivhite" This is confirmed by 
the experiments of Helmholtz ; who, however, has succeeded better with some 
pairs of colours than with others. 

In. my experiments on the spectrum, I came to the same result ; but It 
appeared to me that the very peculiar appearance of the neutral tints produced 



152 EXPERIMENTS ON COLOUR, AS PERCEIVED BY THE EYE. 

was owing to some opticjal effect taking place in the transparent part of the 
eye on the mixture of two rays of very different refrangibility. Most eyes are 
by no means achromatic, so that the images of objects illuminated with mixed 
light of this kind appear divided into two different colours; and even when 
there is no distinct object, the mixtures become in some degree analysed, so as 
to present a very strange, and certainly "anonymous" appearance. 



Additional Note on the more recent experiments of M. Helmholtz*. 

In his former memoir on the Theory of Compound Colours f, M. Helmholtz 
arrived at the conclusion that only one pair of homogeneous colours, orange- 
yellow and indigo-blue, were strictly complementary. This result was shewn by 
Professor Grassmann| to be at variance with Newton's theory of compound 
colours ; and although the reasoning was founded on intuitive rather than 
experimental truths, it pointed out the tests by which Newton's theory must 
be verified or overthrown. In applying these tests, M. Helmholtz made use of 
an apparatus similar to that described by M. Foucault§, by which a screen of 
white paper is illuminated by the mixed light. The field of mixed colour is 
much larger than in M. Helmholtz's former experiments, and the facility of 
forming combinations is much increased. In this memoir the mathematical theory 
of Newton's circle, and of the curve formed by the spectrum, with its possible 
transformations, is completely stated, and the form of this curve is in some 
degree indicated, as far as the determination of the colours which he on oppo- 
site sides of white, and of those which He opposite the part of the curve which 
is wanting. The colours between red and yellow-green are complementary to 
colours between blue-green and violet, and those between yellow-green and blue- 
green have no homogeneous complementaries, but must be neutrahzed by various 
hues of purple, i.e., mixtures of red and violet. The names of the complementary 
colours, with their wave-lengths in air, as deduced from Fraunhofer's measure- 
ments, are given in the following table : — 

• PoggendorflF's Annalen, BA xciv. (I am indebted for the perusal of this Memoir to Professor 
Stokes.) 

+ lb. Bd. Lxxxvii. Annals of Philosophy, 1852, Part ii. 

t Ih. Bd. Lxxxix. Ann. Phil., 1854, April. 

§ lb. Bd. LXixvm. Moigno, Cosmos, 1853, Tom. ii,, p. 232. 



EXPERIMENTS ON COLOUR, AS PERCEIVED BY THE EYE. 



153 



Colour 


Wave-length 


Complementary 
Colour 


Wave-length 


Ratio of 
wave-lengths 


Red ... . 

Orange . . . 
Gold-yellow 
Gold veUow . 
Yellow . . . 
Yellow . . . 
Green-yellow . 


2425 
2244 
2162 
2120 
2095 
2085 
2082 


Green-blue . 
Blue . . . 
Blue . . . 
Blue . . . 
Indigo-blue 
Indigo-blue 
Violet . . 




1818 
1809 
1793 
1781 
1716 
1706 
1600- 


1-334 
1-240 
1-206 
1-190 
1-221 
1-222 
1-301 


(The wave-lengtha are expressed in millionths of a Paris inch.) 



(In order to reduce these wave-lengths to their actual lengths in the eye, 
each must be divided by the index of refraction for that kind of light in the 
medium in which the physical etfect of the vibrations is supposed to take place.) 

Although these experiments are not in themselves sufficient to give the com- 
plete theory of the curve of homogeneous colours, they determine the most 
important element of that theory in a way which seems very accurate, and I 
cannot doubt that when a philosopher who has so fully pointed out the im- 
portance of general theories in physics turns his attention to the theory of 
sensation, he will at least establish the principle that the laws of sensation can 
be successfully investigated only after the corresponding physical laws have been 
ascertained, and that the connection of these two kinds of laws can be appre- 
hended only when the distinction between them is fully recognised. 



Note IV. 



Description of the Figures. Plate I. 

No. 1. is the colour-diagi-am already referred to, representing, cm Newton's principle, the relations of 
diflferent coloured papers to the three standard colours— vermilion, emerald-green, and ultra- 
marine. The initials denoting the colours are explained in the list at page 276, and the 
numbers belonging to them are their coefficients of intensity, the use of which has been 
explained. The initials H.R., H.B., and H.G., represent the red, blue and green papers 
of Mr Hay, and serve to connect this diagram -vith No. (2), which takes these colours for 
its standards. 
VOL. I. 20 



154 EXPEBIMENTS ON COLOUK, AS PERCEIVED BY THE EYE. 

No. 2. represents the relations of Mr Hay's red, blue, green, white, and yellow papers, as deter- 
mined by a large number of experiments at Cambridge. — (See Note II.). The use of the 
point D, in calculating the results of colour-blindness, is explained in the Paper. 

Fig, 3. represents a disc of the larger size, with its slit. 

Fig. 4. shows the mode of combining two discs of the smaller size. 

Fi«^. 5. shows the combination of discs, as placed on the top, in the first experiment described 

in the Paper. 
Fig. 6. represents the method of spinning the top, when speed is required. 
The last four figures are half the actual size. 

Colour-tops of the kind used in these experiments, with paper discs of the colours whose relations 
are represented in No. 1, are to be had of Mr J. M. Bryson, Optician, Edinburgh. 



VOL. I. PLATE L 



Mue,-,^^^ 



^ 



^ 



/ 



C04 



% 



HB06* i 







UCO-4 



^GO-2 / 



^' 



^" 



^^^. 



(1^?1 



tUTyJO 



li^C. 




^6reen/lo 



VOL, L PLATE L 





riG^.4 



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FIG.6 




FIG. 6 



[From the Transactions of the Cambridge Philosophical Society, VoL x. Part i.] 

VIII. On Faraday's Lines of Force. 
[Read Dec. 10, 1855, and Feb. 11, 1856.] 

The present state of electrical science seems peculiarl^^ unfavourable to specu- 
l ation. The laws of the distribution of electricity on the surface of conductors 
have been analytically deduced from experiment; some parts of the mathematical 
theory of magnetism are established, while in other parts the experimental data 
are wanting ; the theory of the conduction of galvan ism and that of the mutual 
attraction of conductors have been reduced to mathematical formulae, but have 
not fallen into relation with the other parts of the science. No electrical theory 
can now be put forth, unless it shews the connexion not only between electricity 
at rest and current electricity, but between the attractions and inductive effects 
of electricity in both states. Such a theory must accurately satisfy those laws, 
the mathematical form of which is known, and must afford the means of calcu- 
lating the effects in the limiting cases where the known formulae are inapplicable. 
In order therefore to appreciate the requirements of the science, the student 
must make himself familiar with a considerable body of most intricate mathe- 
matics, the mer fi retention of which in the memory materially interferes with 
further progress. The first process therefore in the effectual study of the science^, 
must be one of simplification and reduction of the results of previous investiga- 
tion to a form in which the mind can grasp them. The results of this simplifi- 
cation may take the form of a purely mathematical formula or of a physical 
hypothesis. In the first case we entirely lose sight of the phenomena to be 
explained ; and though we may trace out the consequences of given laws, we 
can never obtain more extended views of the connexions of the subject^ If, 
on the other luiml, we adopt a physical hypothesis, we see the phenomena only 
throucrh a medium, and are liable to that blindness to facts and rashness m 

20—2 



156 ON FARADAY S LINES OF FdRCE. 

assumption wKich a partial explanation encourages. "We must therefore discover 
some method of investigation which allows the mind at every step to lay hold 
of a clear physical conception, without being committed to any theory founded 
on the physical science from which that conception is borrowed, so that it is 
neither drawn aside from the subject in pursuit of analytical subtleties, nor carried 
beyond the truth by a favourite hypothesis. 

In order to obtain physical ideas without adopting a physical theory we must 
make ourselves familiar with the existence of physical analogies. By a physical 
analogy I mean that partial similarity between the laws of one science and those 
of another which makes each of them illustrate the other. Thus all the mathe- 
matical sciences are founded on relations between physical laws and laws of 
numbers, so that the aim of exact science is to reduce the problems of nature 
to the determination of quantities by operations with numbers. Passing from 
the most universal of all analogies to a very partial one, we find the same 
resemblance in mathematical form between two different phenomena giving rise 
to a physical theory of light. 

The changes of direction which light undergoes in passing from one medium 
to another, are identical with the deviations of the path of a particle in moving 
through a narrow space in which intense forces act. This analogy, which extends 
only to the direction, and not to the velocity of motion, was long believed to 
he the true explanation of the refraction of Ught ; and we still find it useful 
in the solution of certain problems, in which we employ it without danger, as 
an artificial method. The other analogy, between light and the vibrations of an 
elastic medium, extends much farther, but, though its importance and fruitfulness 
cannot be over-estimated, we must recollect that it is founded only on a resem- 
blance in form between the laws of light and those of vibrations. By stripping 
it of its physical dress and reducing it to a theory of " transverse alternations," 
we might obtain a system of truth strictly founded on observation, but probably 
deficient both in the vividness of its conceptions and the fertility of its method. 
I have said thus much on the disputed questions of Optics, as a preparation 
for the discussion of the almost universally admitted theory of attraction at a 
distance. 

We have all acquired the mathematical conception of these attractions. We 
can reason about them and determine their appropriate forms or formulae. These 
formulae have a distinct mathematical significance, and their results are found 
to be in accordance with natural phenomena. There is no formula in applied 



ON FARADAY'3 lines OF FORCE. 157 

mathematics more consistent with nature than the formula of attractions, and no 
theory better estabUshed in the minds of men than that of the action of bodies 
on one another at a distance. The laws of the conduction of heat in uniform 
media appear at first sight among the most different in their physical relations 
from those relating to attractions. The quantities which enter into them are 
teviperature, flow of heat, conductivity. The word force is foreign to the subject. 
Yet we find that the mathematical laws of the uniform motion of heat in 
homogeneous media are identical in form with those of attractions varying in- 
versely as the square of the distance. We have only to substitute source of 
heat for centre of attrax^tion, flow of heat for accelerating effect of attraction at 
any point, and temperature for potential, and the solution of a problem in 
attractions is transformed into that of a problem in heat. 

This analogy between the formulae of heat and attraction was, I believe, 
first pointed out by Professor William Thomson in the Camh. Math. Journal, 
Vol. III. 

Now the conduction of heat is supposed to proceed by an action between 
contiguous parts of a medium, while the force of attraction is a relation be- 
tween distant bodies, and yet, if we knew nothing more than is expressed in 
the mathematical formulae, there would be nothing to distinguish between the 
one set of phenomena and the other. 

It is true, that if we introduce other considerations and observe additional 
facts, the two subjects will assume very difierent aspects, but the mathematical 
resemblance of some of their laws will remain, and may still be made useful 
in exciting appropriate mathematical ideas. 

It is by the use of analogies of this kind that I have attempted to bring 
before the mind, in a convenient and manageable form, those mathematical ideas 
which are necessary to the study of the phenomena of electricity. The methods 
are generally those suggested by the processes of reasoning which are found in 
the researches of Faraday"*', and which, though they have been interpreted 
mathematically by Prof. Thomson and others, are very generally supposed to be 
of an indefinite and unmathematical character, when compared with those em- 
ployed by the professed mathematicians. By the method which I adopt, I hope 
to render it evident that I am not attempting to estabhsh any physical theory 
of a science in which I have hardly made a single experiment, and that the 
limit of my design is to shew how, by a strict application of the ideas and 

* See especially Series xxxviii. of the Experimental Researcltes, and Phil. Mag. 1852. 



158 ON Faraday's lines of force. 

methods of Faraday, the connexion of the very different orders of phenomena 
which he has discovered may be clearly placed before the mathematical mind. 
I shall therefore avoid as much as I can the introduction of anything which 
does not serve as a direct illustration of Faraday's methods, or of the mathe- 
matical deductions which may be made from them. In treating the simpler 
parts of the subject I shall use Faraday's mathematical methods as well as 
his ideas. When the complexity of the subject requires it, I shall use analytical 
notation, still confining myself to the development of ideas originated by the 
same philosopher. 

I have in the first place to explain and illustrate the idea of "lines of 
force." 

When a body is electrified in any manner, a small body charged with posi- 
tive electricity, and placed in any given position, will experience a force urging 
it in a certain direction. If the small body be now negatively electrified, it will 
be urged by an equal force in a direction exactly opposite. 

The same relations hold between a magnetic body and the north or south 
poles of a small magnet. If the north pole is urged in one direction, the south 
pole is urged in the opposite direction. 

In this way we might find a line passing through any point of space, such 
that it represents the direction of the force acting on a positively electrified 
particle, or on an elementary north pole, and the reverse direction of the force 
on a negatively electrified particle or an elementary south pole. Since at every 
point of space such a direction may be found, if we commence at any point 
and draw a line so that, as we go along it, its direction at any point shall 
always coincide with that of the resultant force at that point, this curve wiU 
indicate the direction of that force for every point through which it passes, and 
might be called on that account a line of force. We might in the same way 
draw other lines of force, till we had filled all space with curves indicating by 
their direction that of the force at any assigned point. 

We should thus obtain a geometrical model of the physical phenomena, 
which would tell us the direction of the force, but we should stiU require some 
method of indicating the intensity of the force at any point. If we consider 
these curves not as mere lines, but as fine tubes of variable section carrying 
an incompressible fluid, then, since the velocity of the fluid is inversely as the 
section of the tube, we may make the velocity vary according to any given law, 
by regulating the section of the tube, and in this way we might represent the 



ON FARADAY S LINES OF FORCE. 



159 



intensity of the force as well as its direction by the motion of the fluid in 
these tubes. This method of representing the intensity of a force by the velocity 
of an imaginary fluid in a tube is applicable to any conceivable system of forces, 
but it is capable of great simplification in the case in which the forces are such 
as can be explained by the hypothesis of attractions varying inversely as the 
square of the distance, such as those observed in electrical and magnetic pheno- 
mena. In the case of a perfectly arbitrary system of forces, there will generally 
be interstices between the tubes ; but in the case of electric and magnetic forces 
it is possible to arrange the tubes so as to leave no interstices. The tubes will 
then be mere surfaces, directing the motion of a fluid filling up the whole space. 
It has been usual to commence the investigation of the laws of these forces by 
at once assuming that the phenomena are due to attractive or repulsive forces 
acting between certain points. We may however obtain a different view of the 
subject, and one more suited to our more difficult inquiries, by adopting for the 
definition of the forces of which we treat, that they may be represented in 
magnitude and direction by the uniform motion of an incompressible fluid. 

I propose, then, first to describe a method by which the motion of such a 
fluid can be clearly conceived; secondly to trace the consequences of assuming 
certain conditions of motion, and to point out the application of the method to 
some of the less complicated phenomena of electricity, magnetism, and galvanism ; 
and lastly to shew how by an extension of these methods, and the introduction 
of another idea due to Faraday, the laws of the attractions and inductive actions 
of magnets and currents may be clearly conceived, without making any assump- 
tions as to the physical nature of electricity, or adding anything to that which 
has been already proved by experiment. 

By referring everything to the purely geometrical idea of the motion of an 
imaginary fluid, I hope to attain generahty and precision, and to avoid the 
dangers arising from a premature theory professing to explain the cause of the 
phenomena. If the results of mere speculation which I have collected are found 
to be of any use to experimental philosophers, in arranging and interpreting 
their results, they will have served their purpose, and a mature theory, in which 
physical facts will be physically explained, will be formed by those who by 
interrogating Nature herself can obtain the only true solution of the questions 
which the mathematical theory suggests. 



160 ON FARADAY S LINES OF FORCE. 



I. Theoi-y of the Motion of an incompressible Fluid. 

(1) The substance here treated of must not be assumed to possess any of 
the properties of ordinary fluids except those of freedom of motion and resistance 
to compression. It is not even a hypothetical fluid which is introduced to 
explain actual phenomena. It is merely a collection of imaginary properties 
which may be employed for establishing certain theorems in pure mathematics in 
a way more intelligible to many minds and more applicable to physical problems 
than that in which algebraic symbols alone are used. The use of the word 
"Fluid" will not lead us into error, if we remember that it denotes a purely 
imaginary substance with the following property : 

The poHion of fluid which at any iTistant occupied a given volume, will at 
any succeeding instant occupy an equal volume. 

This law expresses the incompressibility of the fluid, and furnishes us with 
a convenient measure of its quantity, namely its volume. The unit of quantity 
of the fluid will therefore be the unit of volume. 

(2) The direction of motion of the fluid will in general be dlflerent at 
different points of the space which it occupies, but since the direction is deter- 
minate for every such point, we may conceive a line to begin at any point and 
to be continued so that every element of the line indicates by its direction the 
direction of motion at that point of space. Lines drawn in such a manner that 
their direction always indicates the direction of fluid motion are called lines of 
fluid motion. 

If the motion of the fluid be what is called steady motion, that is, if the 
direction and velocity of the motion at any fixed point be independent of the 
time, these curves will represent the paths of individual particles of the fluid, 
but if the motion be variable this will not generally be the case. The cases 
of motion which will come under our notice will be those of steady motion. 

(3) If upon any surface which cuts the lines of fluid motion we draw a 
closed curve, and if from every point of this curve we draw a line of motion, 
these lines of motion will generate a tubular surface which we may call a tube 
of fluid motion. Since this surface is generated by lines in the direction of fluid 



ON Faraday's lines of force. 161 

motion no part of the fluid can flow across it, so that this imaginary surface 
is as impermeable to the fluid as a real tube. 

(4) The quantity of fluid which in unit of time crosses any fixed section 
of the tube is the same at whatever part of the tube the section be taken. 
For the fluid is incompressible, and no part runs through the sides of the tube, 
therefore the quantity which escapes from the second section is equal to that 
which enters through the first. 

If the tube be such that unit of volume passes through any section in 
unit of time it is called a unit tube of fluid motion. 

(5) In what follows, various units will be referred to, and a finite number 
of lines or surfaces will be drawn, representing in terms of those units the 
motion of the fluid. Now in order to define the motion in every part of the 
fluid, an infinite number of lines would have to be drawn at indefinitely small 
intervals ; but since the description of such a system of lines would involve 
continual reference to the theory of limits, it has been thought better to suppose 
the lines drawn at intervals depending on the assumed unit, and afterwards to 
assume the unit as small as we please by taking a small submultiple of the 
standard unit. 

(6) To define the motion of the whole fluid by means of a system of unit 
tubes. 

Take any fixed surface which cuts all the lines of fluid motion, and draw 
upon it any system of curves not intersecting one another. On the same surface 
draw a second system of curves intersecting the first system, and so arranged 
that the quantity of fluid which crosses the surface within each of the quadri- 
laterals formed by the intersection of the two systems of curves shall be unity 
in unit of time. From every point in a curve of the first system let a line 
of fluid motion be drawn. These lines will form a surface through which no 
fluid passes. Similar impermeable surfaces may be drawn for all the curves of 
the first system. The curves of the second system will give rise to a second 
system of impermeable surfaces, which, by their intersection with the first system, 
will form quadrilateral tubes, which will be tubes of fluid motion. Since each 
quadrilateral of the cutting surface transmits unity of fluid in unity of time, 
every tube in the system will transmit unity of fluid through any of its sections 
in unit of time. The motion of the fluid at every part of the space it occupies 

VOL, I. 21 



162 ON FARADAY S LINES OF FORCE. 

is determined by this system of unit tubes ; for the direction of motion is that 
of the tube through the point in question, and the velocity is the reciprocal 
of the area of the section of the unit tube at. that point. 

(7) We have now obtained a geometrical construction which completely 
defines the motion of the fluid by dividing the space it occupies into a system 
of unit tubes. We have next to shew how by means of these tubes we may 
ascertain various points relating to the motion of the fluid. 

A unit tube may either return into itself, or may begin and end at differ- 
ent points, and these may be either in the boundary of the space in which we 
investigate the motion, or within that space. In the first case there is a con- 
tinual circulation of fluid in the tube, in the second the fluid enters at one end 
and flows out at the other. If the extremities of the tube are in the bound- 
ing surface, the fluid may be supposed to be continually supplied from without 
from an unknown source, and to flow out at the other into an unknown reser- 
voir ; but if the origin of the tube or its termination be within the space under 
consideration, then we must conceive the fluid to be supplied by a source within 
that space, capable of creating and emitting unity of fluid in unity of time, and 
to be afterwards swallowed up by a sink capable of receiving and destroying 
the same amount continually. 

There is nothing self-contradictory in the conception of these sources where 
the fluid is created, and sinks where it is annihilated. The properties of the 
fluid are at our disposal, we have made it incompressible, and now we suppose 
it produced from nothing at certain points and reduced to nothing at others. 
The places of production will be called sources, and their numerical value will be 
the number of units of fluid which they produce in unit of time. The places 
of reduction will, for want of a better name, be called sinks, and will be esti- 
mated by the number of units of fluid absorbed in unit of time. Both places 
win sometimes be called sources, a source being understood to be a sink when 
its sign is negative. 

(8) It is evident that the amount of fluid which passes any fixed surface 
is measured by the number of unit tubes which cut it, and the direction in 
which the fluid passes is determined by that of its motion in the tubes. If 
the surface be a closed one, then any tube whose terminations lie on the same 
side of the surface must cross the surface as many times in the one direction 
as in the other, and therefore must cany as much fluid out of the surface as 



ON Faraday's lines of force. 163 

it carries in. A tube which begins within the surface and ends without it 
will carry out unity of fluid; and one which enters the surface and terminates 
within it will carry in the same quantity. In order therefore to estimate the 
amount of fluid which flows out of the closed surface, we must subtract the 
number of tubes which end within the surface from the number of tubes which 
begin there. If the result is negative the fluid will on the whole flow inwards. 

If we call the beginning of a unit tube a unit source, and its termination 
a unit sink, then the quantity of fluid produced within the surface is estimated 
by the number of unit sources minus the number of unit sinks, and this must 
flow out of the surface on account of the incompressibility of the fluid. 

In speaking of these imit tubes, sources and sinks, we must remember what 
was stated in (5) as to the magnitude of the unit, and how by diminishing 
their size and increasing their number we may distribute them according to any 
law however complicated. 

(9) If we know the direction and velocity of the fluid at any point in 
two diSerent cases, and if we conceive a third case in which the direction and 
velocity of the fluid at any point is the resultant of the velocities in the two 
former cases at corresponding points, then the amount of fluid which passes a 
given fixed surface in the third case will be the algebraic sum of the quantities 
which pass the same surface in the two former cases. For the rate at which 
the fluid crosses any surface is the resolved part of the velocity normal to the 
surface, and the resolved part of the resultant is equal to the sum of the 
resolved parts of the components. 

Hence the number of unit tubes which cross the surface outwards in the 
third case must be the algebraical sum of the numbers which cross it in the 
two former cases, and the number of sources within any closed surface will be 
the sum of the numbers in the two former cases. Since the closed surface may 
be taken as small as we please, it is evident that the distribution of sources 
and sinks in the third case arises from the simple superposition of the distri- 
butions in the two former cases. 



n. TTieory of the uniform motion of an imponderable incompressible fluid 
through a resisting medium. 

(10) The fluid is here supposed to have no inertia, and its motion is opposed 
by the action of a force which we may conceive to be due to the resistance of a 

21—2 



164 ON FARADAY S LINES OF FORCK 

medium through which the fluid is supposed to flow. This resistance depends on 
the nature of the medium, and will in general depend on the direction in which 
the fluid moves, as well as on its velocity. For the present we may restrict 
ourselves to the case of a uniform medium, whose resistance is the same in all 
directions. The law which we assume is as follows. 

Any portion of the fluid moving through the resisting medium is directly 
opposed by a retarding force proportional to its velocity. 

If the velocity be represented by i', then the resistance will be a force equal 
to kv acting on unit of volume of the fluid in a direction contrary to that of 
motion. In order, therefore, that the velocity may be kept up, there must be a 
greater pressure behind any portion of the fluid than there is in front of it, so 
that the difference of pressures may neutrahse the effect of the resistance. Con- 
ceive a cubical unit of fluid (which we may make as small as we please, by (5)), 
and let it move in a direction perpendicular to two of its faces. Then the resist- 
ance will be kv, and therefore the difference of pressures on the first and second 
faces is kv, so that the pressure diminishes in the direction of motion at the rate 
of kv for every unit of length measured along the line of motion ; so that if w6 
measure a length equal to h units, the difference of pressure at its extremities 
will be kvh. 

(11) Since the pressure is supposed to vary continuously in the fluid, all 
the points at which the pressure is equal to a given pressure p will lie on a 
certain surface which we may call the surface (p) of equal pressure. If a series 
of these surfaces be constructed in the fluid corresponding to the pressures 0, 1, 
2, 3 &c., then the number of the surface will indicate the pressure belonging to 
it, and the surface may be referred to as the surface 0, 1, 2 or 3. The unit of 
pressure is that pressure which is produced by unit of force acting on unit of 
surface. In order therefore to diminish the unit of pressure as in (5) we must 
diminish the unit of force in the same proportion. 

(12) It is easy to see that these surfaces of equal pressure must be perpen- 
dicular to the lines of fluid motion; for if the fluid were to move in any other 
direction, there would be a resistance to its motion which could not be balanced 
by any difference of pressures. (We must remember that the fluid here con- 
sidered has no inertia or mass, and that its properties are those only which are 
formally assigned to it, so that the resistances and pressures are the only things 



ON Faraday's lines of force. 165 

to be considered.) There are therefore two sets of surfaces which by their inter- 
section form the system of unit tubes, and the system of surfaces of equal pres- 
sure cuts both the others at right angles. Let h be the distance between two 
consecutive surfaces of equal pressure measured along a line of motion, then since 
the difference of pressures = 1, 

kvh= 1, 

which determines the relation of v to h, so that one can be found when the 
other is known. Let s be the sectional area of a unit tube measured on a 
surface of equal pressure, then since by the definition of a unit tube 

vs = \, 

we find by the last equation 

s = kh. 

(13) The surfaces of equal pressure cut the unit tubes into portions whose 
length is h and section s. These elementary portions of unit tubes will be called 
unit cells. In each of them unity of volume of fluid passes from a pressure p to 
a pressure (p — 1) in unit of time, and therefore overcomes unity of resistance in 
that time. The work spent in overcoming resistance is therefore unity in every 
cell in every unit of time. 

(14) If the surfaces of equal pressure are known, the direction and magni- 
tude of the velocity of the fluid at any point may be found, after which the 
complete system of unit tubes may be constructed, and the beginnings and end- 
ings of these tubes ascertained and marked out as the sources whence the fluid 
is derived, and the sinks where it disappears. In order to prove the converse of 
this, that if the distribution of sources be given, the pressure at every point may 
be found, we must lay down certain preliminary propositions. 

(15) If we know the pressures at every point in the fluid in two different 
cases, and if we take a third case in which the pressure at any point is the 
sum of the pressures at corresponding points in the two former cases, then the 
velocity at any point in the third case is the resultant of the velocities in the 
other two, and the distribution of sources is that due to the simple superposition 
of the sources in the two former cases. 

For the velocity in any direction is proportional to the rate of decrease of 
the pressure in that direction; so that if two systems of pressures be added 



166 ON FARADAY S LINES OF FORCE. 

together, since the rate of decrease of pressure along any line will be the sum 
of the combined rates, the velocity in the new system resolved in the same 
direction will be the sum of the resolved parts in the two original systems. 
The velocity in the new system will therefore be th€ resultant of the velocities 
at corresponding points in the two former systems. 

It follows from this, by (9), that the (quantity of fluid which crosses any 
fixed surface is, in the new system, the sum of the corresponding quantities in 
the old ones, and that the sources of the two original systems are simply 
combined to form the third. 

It is evident that in the system in which the pressure is the diiBPerence 
of pressure in the two given systems the distribution of sources will be got 
by changing the sign of all the sources in the second system and adding them 
to those in the first. 

(16) If the pressure at every point of a closed surface be the same and 
equal to p, and if there be no sources or sinks within the surface, then there 
will be no motion of the fluid within the surface, and the pressure within it 
will be uniform and equal to p. 

For if there be motion of the fluid within the surface there will be tubes 
of fluid motion, and these tubes must either return into themselves or be 
terminated either within the surface or at its boundary. Now since the fluid 
always flows from places of greater pressure to places of less pressure, it 
cannot flow in a re-entering curve; since there are no sources or sinks within 
the surface, the tubes cannot begin or end except on the surface ; and since 
the pressure at all points of the surface is the same, there can be no motion 
in tubes having both extremities on the surface. Hence there is no motion 
within the surface, and therefore no difference of pressure which would cause 
motion, and since the pressure at the bounding surface is p, the pressure at 
any point within it is also p. 

(17) If the pressure at every point of a given closed surface be known, 
and the distribution of sources within the surface be also known, then only 
one distribution of pressures can exist within the surface. 

For if two different distributions of pressures satisfying these conditions 
could be found, a third distribution could be formed in which the pressure at 
any point should be the difference of the pressures in the two former distri- 
butions. In this case, since the pressures at the surface and the sources within 



ON Faraday's lines of force. 107 

it are the same in both distributions, the pressure at the surface in the third 
distribution would be zero, and all the sources within the surface would 
vanish, by (15). 

Then by (16) the pressure at every point in the third distribution must 
be zero ; but this is the difference of the pressures in the two former cases, 
and therefore these cases are the same, and there is only one distribution of 
pressure possible. 

(18) Let us next determine the pressure at any point of an infinite body 
of fluid in the centre of which a unit source is placed, the pressure at an 
infinite distance from the source being supposed to be zero. 

The fluid will flow out from the centre symmetrically, and since unity of 
volume flows out of every spherical surface surrounding the point in unit of 
time, the velocity at a distance r from the source will be 

k 

The rate of decrease of pressure is therefore hv or — -^, and since the 

pressure = when r is infinite, the actual pressure at any point will be 

= A 

The pressure is therefore inversely proportional to the distance from the 
source. 

It is evident that the pressure due to a unit sink will be negative and 

equal to — - — . 

If we have a source formed by the coalition of »S' unit sources, then the 

TcS 
resulting pressure will be X>=t—,, so that the pressure at a given distance 

varies as the resistance and number of sources conjointly. 

(19) If a number of sources and sinks coexist in the fluid, then in order 
to determine the resultant pressure we have only to add the pressures which 
each source or sink produces. For by (15) this will be a solution of the 
problem, and by (17) it will be the only one. By this method we can 
determine the pressures due to any distribution of sources, as by the method 



168 ON Faraday's lines of forck 

of (14) we can determine the distribution of sources to which a given distri- 
bution of pressures is due. 

(20) We have next to shew that if we conceive any imaginary surface 
as fixed in space and intersecting the lines of motion of the fluid, we may 
substitute for the fluid on one side of this surface a distribution of sources 
upon the surface itself without altering in any way the motion of the fluid 
on the other side of the surface. 

For if we describe the system of unit tubes which defines the motion of 
the fluid, and wherever a tube enters through the surface place a unit source, 
and wherever a tube goes out through the surface place a unit sink, and at the 
same time render the surface impermeable to the fluid, the motion of the fluid 
in the tubes will go on as before. 

(21) If the system of pressures and the distribution of sources which pro- 
duce them be known in a medium whose resistance is measured by k, then in 
order to produce the same system of pressures in a medium whose resistance 
is unity, the rate of production at each source must be multiplied by k. For 
the pressure at any point due to a given source varies as the rate of produc- 
tion and the resistance conjointly; therefore if the pressure be constant, the 
rate of production must vary inversely as the resistance. 

(22) On the conditions to he fulfilled at a surface which separates two media 
whose coefficients of resistance are k and k\ 

These are found from the consideration, that the quantity of fluid which 
flows out of the one medium at any point flows into the other, and that the 
pressure varies continuously from one medium to the other. The velocity normal 
to the surface is the same in both media, and therefore the rate of diminution 
of pressure is proportional to the resistance. The direction of the tubes of 
motion and the surfaces of equal pressure will be altered after passing through 
the surface, and the law of this refraction will be, that it takes place in the 
plane passing through the direction of incidence and the normal to the surface, 
and that the tangent of the angle of incidence is to the tangent of the angle 
of refraction as k' is to k. 

(23) Let the space within a given closed surface be filled with a medium 
different from that exterior to it, and let the pressures at any point of this 
compound system due to a given distribution of sources within and without 



ON fakaday's lines of force. 169 

the surface be given ; it is required to determine a distribution of sources which 
would produce the same system of pressures in a medium whose coefficient of 
resistance is unity. 

Construct the tubes of fluid motion, and wherever a unit tube enters either 
medium place a unit source, and wherever it leaves it place a unit sink. Then 
if we make the surface impermeable all will go on as before. 

Let the resistance of the exterior medium be measured by k, and that of 
the interior by V. Then if we multiply the rate of production of all the sources 
in the exterior medium (including those in the surface), by k, and make the 
coefficient of resistance unity, the pressures will remain as before, and the same 
will be true of the interior medium if we multiply all the sources in it by k', 
including those in the surface, and make its resistance unity. 

Since the pressures on both sides of the surface are now equal, we may 
suppose it permeable if we please. 

We have now the original system of pressures produced in a uniform medium 
by a combination of three systems of sources. The first of these is the given 
external system multipHed by k, the second is the given internal system multi- 
plied by k', and the third is the system of sources and sinks on the surface 
itself. In the original case every source in the external medium had an equal 
sink in the internal medium on the other side of the surface, but now the 
source is multiplied by k and the sink by k', so that the result is for every 
external unit source on the surface, a source ={k — k'). By means of these three 
systems of sources the original system of pressures may be produced in a medium 
for which k = \. 

(24) Let there be no resistance in the medium within the closed surface, 
that is, let /t' = 0, then the pressure within the closed surface is uniform and 
equal to p, and the pressure at the surface itself is also p. If by assuming 
any distribution of pairs of sources and sinks within the surface in addition to 
the given external and internal sources, and by supposing the medium the same 
within and without the surface, we can render the pressure at the surface uni- 
form, the pressures so found for the external medium, together with the uniform 
pressure p in the internal medium, will be the true and only distribution of 
pressures which is possible. 

For if two such distributions could be found by taking diffijrent imaginary 
distributions of pairs of sources and sinks within the medium, then by taking 
VOL. I. 22 



170 ON Faraday's lines of foece. 

the difference of the two for a third distribution, we should have the pressure 
of the bounding surface constant in the new system and as many sources as 
sinks within it, and therefore whatever fluid flows in at any point of the surface, 
an equal quantity must flow out at some other point. 

In the external medium all the sources destroy one another, and we have 
an infinite medium without sources surrounding the internal medium. The pres- 
sure at infinity is zero, that at the surface is constant. If the pressure at the 
surface is positive, the motion of the fluid must be outwards from every point 
of the surface ; if it be negative, it must flow inwards towards the surface. But 
it has been shewn that neither of these cases is possible, because if any fluid 
enters the surface an equal quantity must escape, and therefore the pressure at 
the surface is zero in the third system. 

The pressure at all points in the boundary of the internal medium in the 
third case is therefore zero, and there are no sources, and therefore the pressure 
is everywhere zero, by (16). 

The pressure in the bounding surface of the internal medium is also zero, 
and there is no resistance, therefore it is zero throughout; but the pressure in 
the third case is the difference of pressures in the two given cases, therefore 
these are equal, and there is only one distribution of pressure which is possible, 
namely, that due to the imaginary distribution of sources and sinks. 

(25) When the resistance is infinite in the internal medium, there can be 
no passage of fluid through it or into it. The bounding surface may therefore 
be considered as impermeable to the fluid, and the tubes of fluid motion will 
run along it without cutting it. 

If by assuming any arbitrary distribution of sources within the surface in 
addition to the given sources in the outer medium, and by calculating the 
resulting pressures and velocities as in the case of a uniform medium, we can 
fulfil the condition of there being no velocity across the surface, the system of 
pressures in the outer medium will be the true one. For since no fluid passes 
through the surface, the tubes in the interior are independent of those outside, 
and may be taken away without altering the external motion. 

(26) If the extent of the internal medium be small, and if the difference 
of resistance in the two media be also small, then the position of the unit tubes 
will not be much altered from what it would be if the external medium filled 
the whole space. 



ON FARADAY S LINES OF FORCE. 171 

Oq this supposition we can easily calculate the kind of alteration which 
the introduction of the internal medium will produce ; for wherever a unit tube 

enters the surface we must conceive a source producing fluid at a rate -^^ , 

and wherever a tube leaves it we must place a sink annihilating fluid at the 

k'-k 
rate — ^ , then calculating pressures on the supposition that the resistance in 

both media is k, the same as in the external medium, we shall obtain the true 
distribution of pressures very approximately, and we may get a better result 
by repeating the process on the system of pressures thus obtained. 

(27) If instead of an abrupt change from one coeflBcient of resistance to 
another we take a case in which the resistance varies continuously from point 
to point, we may treat the medium as if it were composed of thin shells each 
of which has uniform resistance. By properly assuming a distribution of sources 
over the surfaces of separation of the shells, we may treat the case as if the 
resistance were equal to unity throughout, as in (23). The sources will then 
be distributed continuously throughout the whole medium, and will be positive 
whenever the motion is from places of less to places of greater resistance, and 
negative when in the contrary direction. 

(28) Hitherto we have supposed the resistance at a given point of the 
medium to be the same in whatever direction the motion of the fluid takes 
place ; but we may conceive a case in which the resistance is different in 
different directions. In such cases the lines of motion will not in general be 
perpendicular to the surfaces of equal pressure. If a, 6, c be the components 
of the velocity at any point, and a, yS, y the components of the resistance at 
the same point, these quantities will be connected by the following system of 
linear equations, which may be called ''equations of conduction" and will be 
referred to by that name. 

a^P,a + QS + R.y, 

h = Fj3+Q,y + EA, 

c = P,y+Q,a + JR,l3. 

In these equations there are nine independent coefficients of conductivity. In 

order to simplify the equations, let us put 

Qt + Ji, = 2S„ Q,-B, = 2lT, 

&c &c. 

22—2 



172 ON Faraday's lines of force. 

where 4^ = «?,-i2,)' + (^»-^.)' + (^3-^s)', 

and I, m, n are direction-cosines of a certain fixed line in space. 

The equations then become 

a = P,a + SJ3 + S,y + (nfi -my) T, 
b=F^ + S,y + S,a + {lY - na) T, 
c = P,y + S,a + S^ + {ma~ l^) T. 

By the ordinary transformation of co-ordinates we may get rid of the 
coeflBcients marked S. The equations then become 

a = P(a + (n'^-m'y)T, 

b = P:/3 + {ry-n'a)T, 

c = P,y+{m'a- Vfi) T, 
where I', m, n' are the direction-cosines of the fixed line with reference to the 
new axes. If we make 



the equation of continuity 



becomes 



%^-i' -^-|. 



da dh c^c _ 
dx dy dz ' 



' dx'^ ' dy'^^' dz' ^' 



and if we make x = JP^^, y^^fPT^], z = JP^l, 

^'^^■^ 3|+^ + ? = °- 

the ordinary equation of conduction. 

It appears therefore that the distribution of pressures is not altered by 
the existence of the coefficient T. Professor Thomson has shewn how to 
conceive a substance in which this coefficient determines a property having 
reference to an axis, which unlike the axes of P^, P^, P^ is dipolar. 

For further information on the equations of conduction, see Professor 
Stokes On the Conduction of Heat in Crystals {Cambridge and Dublin Math. 
Journ.), and Professor Thomson On the Dynamical Theory of Heat, Part v. 
{Transactions of Royal Society of Edinburgh, VoL xxi. Part i.). 



ON Faraday's lines of force. 173 

It is evident that all that has been proved in (14), (15), (16), (17), with 
respect to the superposition of different distributions of pressure, and there being 
only one distribution of pressures corresponding to a given distribution of sources, 
will be true also in the case in which the resistance varies from point to point, 
and the resistance at the same point is different in different directions. For 
il' we examine the proof we shall find it applicable to such cases as well as to 
that of a uniform medium. 

(29) We now are prepared to prove certain general propositions which are 
true in the most general case of a medium whose resistance is different in 
different directions and varies from point to point. 

We may by the method of (28), when the distribution of pressures is 
known, construct the surfaces of equal pressure, the tubes of fluid motion, and 
the sources and sinks. It is evident that since in each cell into which a unit 
tube is divided by the surfaces of equal pressure unity of fluid passes from 
pressure p to pressure (p — 1) in unit of time, unity of work is done by the 
fluid in each cell in overcoming resistance. 

The number of cells in each unit tube is determined by the number of 
surfaces of equal pressure through which it passes. If the pressure at the 
beginning of the tube be p and at the end p\, then the number of cells in 
it will be p—p- Now if the tube had extended from the source to a place 
where the pressure is zero, the number of cells would have been p, and if 
the tube had come from the sink to zero, the number would have been p\ 
and the true number is the difference of these. 

Therefore if we find the pressure at a source S from which S tubes 
proceed to be p, Sp \s. the number of cells due to the source S ; but if iS' of 
the tubes terminate in a sink at a pressure p\ then we must cut off S p cells 
from the number previously obtained. Now if we denote the source of S 
tubes by S, the sink of S tubes may be written -S, sinks always being 
reckoned negative, and the general expression for the number of cells in the 
system will be S (5p). 

(30) The same conclusion may be arrived at by observing that unity of 
work is done on each cell. Now in each source S, S units of fluid are 
expelled against a pressure p, so that the work done by the fluid in over- 
coming resistance is Sj?. At each sink in which S' tubes terminate, S' units 
of fluid sink into nothing under pressure p' ; the work done upon the fluid by 



174 ON Faraday's lines of force. 

the pressure is therefore S' p\ The whole work done by the fluid may there- 
fore be expressed by 

W = tSp^tS'p, 
or more concisely, considering sinks as negative sources, 

W = t(Sp). 

(31) Let S represent the rate of production of a source in any medium, 
and let p be the pressure at any given point due to that source. Then if we 
superpose on this another equal source, every pressure will be doubled, and 
thus by successive superposition we find that a source nS would produce a 
pressure np, or more generally the pressure at any point due to a given 
source varies as the rate of production of the source. This may be expressed 
by the equation 

p = RS, 

where R is a, coefficient depending on the nature of the medium and on the 
positions of the source and the given point. In a uniform medium whose 
resistance is measured by k, 

R may be called the coefficient of resistance of the medium between the source 
and the given point. By combining any number of sources we have generally 

p = %{RS), 

(32) In a uniform medium the pressure due to a source S 

k S 

At another source S' at a distance r we shall have 

a, k SS' CI f 

if 2^' he the pressure at S due to S\ If therefore there be two systems of 
sources X{S) and %{S'), and if the pressures due to the first be p and to the 
second p', then 

2(S» = 2{S/). 

For every term S'p has a term Sp' equal to it. 



ON Faraday's lines of force. 175 

(33) Suppose that in a uniform medium the motion of the fluid is every- 
where parallel to one plane, then the surfaces of equal pressure will be 
perpendicular to this plane. If we take two parallel planes at a distance equal 
to k from each other, we can divide the space between these planes into unit 
tubes by means of cylindric surfaces perpendicular to the planes, and these 
together with the surfaces of equal pressure will divide the space into cells of 
which the length is equal to the breadth. For if h be the distance between 
consecutive surfaces of equal pressure and s the section of the unit tube, we 
have by (13) s = kh. 

But s is the product of the breadth and depth ; but the depth is k, 
therefore the breadth is h and equal to the length. 

If two systems of plane curves cut each other at right angles so as to 
divide the plane into little areas of which the length and breadth are equal, 
then by taking another plane at distance k from the first and erecting 
cyhndric surfaces on the plane curves as bases, a system of cells will be 
formed which will satisfy the conditions whether we suppose the fluid to run 
along the first set of cutting lines or the second*. 



Application of the Idea of Lines of Force. 

I have now to shew how the idea of lines of fluid motion as described 
above may be modified so as to be apphcable to the sciences of statical elec- 
tricity, permanent magnetism, magnetism of induction, and uniform galvanic 
currents, reserving the laws of electro-magnetism for special consideration. 

I shall assume that the phenomena of statical electricity have been ah*eady 
explained by the mutual action of two opposite kinds of matter. If we consider 
one of these as positive electricity and the other as negative, then any two 
particles of electricity repel one another with a force which is measured by the 
product of the masses of the particles divided by the square of their distance. 

Now we found in (18) that the velocity of our imaginary fluid due to a 
source *S at a distance r varies inversely as r". Let us see what will be the 
effect of substituting such a source for every particle of positive electricity. The 
velocity due to each source would be proportional to the attraction due to the 
corresponding particle, and the resultant velocity due to all the sources would 

* See Cambridge and Dublin MalJiematical Jownal, Vol. in. p. 286. 



176 ON Faraday's lines of force. 

be proportional to the resultant attraction of all the particles. Now we may find 
the resultant pressure at any point by adding the pressures due to the given 
sources, and therefore we may find the resultant velocity in a given direction 
from the rate of decrease of pressure in that direction, and this will be 
proportional to the resultant attraction of the particles resolved in that direction. 
Since the resultant attraction in the electrical problem is proportional to 
the decrease of pressure in the imaginary problem, and since we may select 
any values for the constants in the imaginary problem, we may assume that the 
resultant attraction in any direction is numerically equal to the decrease of 
pressure in that direction, or 

ax 
By this assumption we find that if F be the potential, 
dV=Xdx+ Ydy + Zdz= -dp, 
or since at an infinite distance F= and p = 0, V= —p. 
In the electrical problem we have 



7. Q 

In the fluid p = S [- 



^ r 



S= -jr dm. 



If k be supposed very great, the amount of fluid produced by each source 
in order to keep up the pressures will be very small. 

The potential of any system of electricity on itself will be 

If t (dm), X (dm') be two systems of electrical particles and p, p' the potentials 
due to them respectively, then by (32) 

or the potential of the first system on the second is equal to that of the second 
system on the first. 



ON Faraday's lines of force. 177 

So that in the ordinary electrical problems the analogy in fluid motion is 
of this kind : 

V=-p, 

dm = -— S, 
Ait 

whole potential of a system = -XVdm^— W, where W is the work done by 

the fluid in overcoming resistance. 

The lines of forces are the unit tubes of fluid motion, and they may be 
estimated numerically by those tubes. 



Theory of Dielectrics, 

The electrical induction exercised on a body at a distance depends not 
only on the distribution of electricity in the inductric, and the form and posi- 
tion of the inducteous body, but on the nature of the interposed medium, or 
dielectric. Faraday* expresses this by the conception of one substance having 
a greater inductive capacity, or conducting the lines of inductive action more 
freely than another. If we suppose that in our analogy of a fluid in a resisting 
medium the resistance is diflerent in difierent media, then by making the 
resistance less we obtain the analogue to a dielectric which more easily conducts 
Faraday's lines. 

It is evident from (23) that in this case there will always be a:n apparent 
distribution of electricity on the surface of the dielectric, there being negative 
electricity where the lines enter and positive electricity where they emerge. In 
the case of the fluid there are no real sources on the surface, but we use 
them merely for purposes of calculation. In the dielectric there may be no 
real charge of electricity, but only an apparent electric action due to the surface. 

If the dielectric had been of less conductivity than the surrounding medium, 
we should have had precisely opposite eflects, namely, positive electricity where 
lines enter, and negative where they emerge. 



* Series xi. 
VOL. I. 



23 



178 ON Faraday's lines of force. 

If the conduction of the dielectric is perfect or nearly so for the small 
quantities of electricity with which we have to do, then we have the case of 
(24). The dielectric is then considered as a conductor, its surface is a surface 
of equal potential, and the resultant attraction near the surface itself is per- 
pendicular to it. 

Theory of Permanent Magnets. 

A magnet is conceived to be made up of elementary magnetized particles, 
each of which has its own north and south poles, the action of which upon 
other north and south poles is governed by laws mathematically identical with 
those of electricity. Hence the same application of the idea of lines of force 
can be made to this subject, and the same analogy of fluid motion can be 
employed to illustrate it. 

But it may be useful to examine the way in which the polarity of the 
elements of a magnet may be represented by the unit cells in fluid motion. 
In each unit cell unity of fluid enters by one face and flows out by the opposite 
face, so that the first face becomes a unit sink and the second a unit source 
with respect to the rest of the fluid. It may therefore be compared to an 
elementary magnet, having an equal quantity of north and south magnetic 
matter distributed over two of its faces. If we now consider the cell as forming 
part of a system, the fluid flowing out of one cell will flow into the next, and 
so on, so that the source will be transferred from the end of the cell to the 
end of the unit tube. If all the unit tubes begin and end on the bounding 
surface, the sources and sinks will be distributed entirely on that surface, and in 
the case of a magnet which has what has been called a solenoidal or tubular 
distribution of magnetism, all the imaginary magnetic matter will be on the 
surface^". 

Theory of Paramagnetic and Diamagnetic Induction. 

Faraday t has shewn that the effects of paramagnetic and diamagnetic bodies 
in the magnetic field may be explained by supposing paramagnetic bodies to 

* See Professor Thomson On the Matliematical Theory of Magnetism, Chapters in. and v. Ph^. 
Trans. 1851. 

t Experimental Researches (3292). 



ON FARADAY S LINES OF FORCE. 179 

conduct the lines of force better, and diamagnetic bodies worse, than the 
surrounding medium. Bj referring to (23) and (26), and supposing sources to 
represent north magnetic matter, and sinks south magnetic matter, then if a 
paramagnetic body be in the neighbourhood of a north pole, the lines of force 
on entering it will produce south magnetic matter, and on leaving it they will 
produce an equal amount of north magnetic matter. Since the quantities of 
magnetic matter on the whole are equal, but the southern matter is nearest 
to the north pole, the result will be attraction. If on the other hand the body 
be diamagnetic, or a worse conductor of lines of force than the surrounding 
medium, there will be an imaginary distribution of northern magnetic matter 
where the lines pass into the worse conductor, and of southern where they pass 
out, so that on the whole there will be repulsion. 

"We may obtain a more general law from the consideration that the poten- 
tial of the whole system is proportional to the amount of work done by the 
fluid in overcoming resistance. The introduction of a second medium increases 
or diminishes the work done according as the resistance is greater or less than 
that of the first medium. The amount of this increase or diminution will vary 
as the square of the velocity of the fluid. 

Now, by the theory of potentials, the moving force in any direction is 
measured by the rate of decrease of the potential of the system in passing along 
that direction, therefore when ¥, the resistance within the second medium, is 
greater than k, the resistance in the surrounding medium, there is a force tend- 
ing from places where the resultant force v is greater to where it is less, so 
that a diamagnetic body moves from greater to less values of the resultant 
force *. 

In paramagnetic bodies V is less than k, so that the force is now from 
points of less to points of greater resultant magnetic force. Since these results 
depend only on the relative values of k and k', it is evident that by changing 
the surrounding medium, the behaviour of a body may be changed from para- 
magnetic to diamagnetic at pleasure. 

It is evident that we should obtain the same mathematical results if we 
had supposed that the magnetic force had a power of exciting a polarity in 
bodies which is in the same direction as the lines in paramagnetic bodies, and 



* Experimental Heaearchei (2797), (2798). See Thomson, Canibridge and Dublin Mathe)naticcU 
Journal, May, 1847. 

23—2 



180 ON Faraday's lines of force. 

in the reverse direction in diamagnetic bodies*. ' In fact we have not as yet 
come to any facts which would lead us to choose any one out of these three 
theories, that of lines of force, that of imaginary magnetic matter, and that of 
induced polarity. As the theory of lines of force admits of the most precise, 
and at the same time least theoretic statement, we shall allow it to stand for 
the present. 

TJieory of Magnecrystallic Induction. 

Ihe theory of Faraday t with respect to the behaviour of crystals in the 
magnetic field may be thus stated. In certain crystals and other substances the 
lines of magnetic force are conducted with difierent facility in different directions. 
The body when suspended in a uniform magnetic field will turn or tend to turn 
into such a position that the lines of force shall pass through it with least resist- 
ance. It is not difficult by means of the principles in (28) to express the laws 
of this kind of action, and even to reduce them in certain cases to numerical 
formulae. The principles of induced polarity and of imaginary magnetic matter 
are here of Httle use; but the theory of lines of force is capable of the most 
perfect adaptation to this class of phenomena. 



Theory of the Conduction of Current Electricity. 

It is in the calculation of the laws of constant electric currents that the 
theory of fluid motion which we have laid down admits of the most direct appU- 
cation. In addition to the researches of Ohm on this subject, we have those 
of M. Kirchhoff, Ann. de Chim. xli. 496, and of M. Quincke, XLvn. 203, on the 
Conduction of Electric Currents in Plates. According to the received opinions 
we have here a current of fluid moving uniformly in conducting circuits, which 
oppose a resistance to the current which has to be overcome by the application 
of an electro-motive force at some part of the circuit. On account of this 
resistance to the motion of the fluid the pressure must be diflerent at difierent 
points in the circuit. This pressure, which is commonly called electrical tension, 

♦ Uxp. Ees. (2429), (3320). See Weber, PoggendorflF, lxxxvil p. H5. Prof. TyndaU, Fhxi. 
Trans. 1856, p. 237. 

t Fxp. Res. (2836), &c. 



ON FAHADAYS LINES OF FORCE. 181 

is found to be physically identical with the potential in statical electricity, and 
thus we have the means of connecting the two sets of phenomena. If we knew 
what amount of electricity, measured statically, passes along that current which 
we assume as our unit of current, then the connexion of electricity of tension 
with current electricity would be completed*. This has as yet been done only 
approximately, but we know enough to be certain that the conducting powers of 
diflferent substances differ only in degree, and that the difference between glass 
and metal is, that the resistance is a great but finite quantity in glass, and a 
small but finite quantity in metal. Thus the analogy between statical electricity 
and fluid motion turns out more perfect than we might have supposed, for there 
the induction goes on by conduction just as in current electricity, but the quan- 
tity conducted is insensible owing to the great resistance of the dielectricst. 



On Electro-motive Forces. 

When a uniform current exists in a closed circuit it is evident that some 
other forces must act on the fluid besides the pressures. For if the current 
were due to difference of pressures, then it would flow from the point of 
greatest pressure in both directions to the point of least pressure, whereas in 
reahty it circulates in one direction constantly. We must therefore admit the 
existence of certain forces capable of keeping up a constant current in a closed 
circuit. Of these the most remarkable is that which is produced by chemical 
action. A cell of a voltaic battery, or rather the surface of separation of the 
fluid of the ceU and the zinc, is the seat of an electro-motive force which 
can maintain a current in opposition to the resistance of the circuit. If we 
adopt the usual convention in speaking of electric currents, the positive current 
is from the fluid through the platinum, the conducting circuit, and the zinc, 
back to the fluid again. If the electro-motive force act only in the surface of 
separation of the fluid and zinc, then the tension of electricity in the fluid 
must exceed that in the zinc by a quantity depending on the nature and 
length of the circuit and on the strength of the current in the conductor. 
In order to keep up this difference of pressure there must be an electro-motive 
force whose intensity is measured by that difference of pressure. If F be the 
electro-motive force, / the quantity of the current or the number of electrical 

♦ See Exp. Ees. (371). t Hxp. Ret. Vol iii. p. 513. 



182 ON Faraday's lines of force. 

units delivered in unit of time, and K a quEfntity depending on the length 
and resistance of the conducting circuit, then 

F=IK=p-p\ 
where p is the electric tension in the fluid and p' in the zinc. 

If the circuit be broken at any point, then since there is no current the 
tension of the part which remains attached to the platinum will be p, and 
that of the other will be p, p-p or F afibrds a measure of the intensity 
of the current. This distinction of quantity and intensity is very useful *, 
but must be distinctly understood to mean nothing more than this : — The 
quantity of a current is the amount of electricity which it transmits in unit 
of time, and is measured by / the number of unit currents which it contains. 
The intensity of a current is its power of overcoming resistance, and is 
measured by F or IK, where K is the resistance of the wliole circuit. 

The same idea of quantity and intensity may be applied to the case of 

magnetism f. The quantity of magnetization in any section of a magnetic 

body is measured by the number of lines of magnetic force which pass through 

it. The intensity of magnetization in the section depends on the resisting 

power of the section, as well as on the number of lines which pass through 

it. If h be the resisting power of the material, and S the area of the section, 

and / the number of lines of force which pass through it, then the whole 

intensity throughout the section 

h 
= F=I- 

When magnetization is produced by the influence of other magnets only, 
we may put p for the magnetic tension at any point, then for the whole 
magnetic solenoid 



F=l(^dx = IK=p-p, 



When a solenoidal magnetized circuit returns into itself, the magnetization 
does not depend on difference of tensions only, but on some magnetizing force 
of which the intensity is F. 

If i be the quantity of the magnetization at any point, or the number of 
lines of force passing through unit of area in the section of the solenoid, then 

* Hxp. Res. Vol. HI. p. 519. t Exp. Res. (2870), (3293). 



ON Faraday's lines of force. 183 

the total quantity of magnetization in the circuit is the number of lines which 
pass through any section, I=Xidydz, where dydz is the element of the section, 
and the summation is performed over the whole section. 

The intensity of magnetization at any point, or the force required to 
keep up the magnetization, is measured by Jci=f, and the total intensity of 
magnetization in the circuit is measured by the sum of the local intensities all 
round the circuit, 

F=t(fdx), 

where dx is the element of length in the circuit, and the summation is extended 
round the entire circuit. 

In the same circuit we have always F = IK, where K is the total resistance 
of the circuit, and depends on its form and the matter of which it is 
composed. 



On the Action of closed Currents at a Distance. 

The mathematical laws of the attractions and repulsions of conductors have 
been most ably investigated by Ampere, and his results have stood the test of 
subsequent experiments. 

From the single assumption, that the action of an element of one current 
upon an element of another current is an attractive or repulsive force acting 
in the direction of the line joining the two elements, he has determined by 
the simplest experiments the mathematical form of the law of attraction, and 
has put this law into several most elegant and useful forms. We must 
recollect however that no experiments have been made on these elements of 
currents except under the form of closed currents either in rigid conductors 
or in fluids, and that the laws of closed currents can only be deduced from 
such experiments. Hence if Ampere's formulae applied to closed currents give 
true results, their truth is not proved for elements of currents unless we 
assume that the action between two such elements must be along the line which 
joms them. Although this assumption is most warrantable and philosophical in 
the present state of science, it wiQ be more conducive to freedom of investi- 
gation if we endeavour to do without it, and to assume the laws of closed currents 
as the ultimate datum of experiment. 



384 ON fahaday's lines of force. 

Ampere has shewn that when currents are combined according to the law 
of the parallelogram of forces, the force due to the resultant current is the 
resultant of the forces due to the component currents, and that equal and 
opposite currents generate equal and opposite forces, and when combined 
neutralize each other. 

He has also shewn that a closed circuit of any form has no tendency to 
turn a moveable circular conductor about a fixed axis through the centre of 
the circle perpendicular to its plane, and that therefore the forces in the case 
of a closed circuit render Xdx + Ydy + Zdz a complete differential. 

Finally, he has shewn that if there be two systems of circuits similar 
and similarly situated, the quantity of electrical current in corresponding 
conductors being the same, the resultant forces are equal, whatever be the 
absolute dimensions of the systems, which proves that the forces are, cceteris 
paribus, inversely as the square of the distance. 

From these results it follows that the mutual action of two closed currents 
whose areas are very small is the same as that of two elementary magnetic 
bars magnetized perpendicularly to the plane of the currents. 

The direction of magnetization of the equivalent magnet may be pre- 
dicted by remembering that a current travelling round the earth from east 
to west as the sun appears to do, would be equivalent to that magnetization 
which the earth actually possesses, and therefore in the reverse direction to 
that of a magnetic needle when pointing freely. 

If a number of closed unit currents in contact exist on a surface, then at 
aU points in which two currents are in contact there will be two equal and 
opposite currents which will produce no effect, but all round the boundary of the 
surfeice occupied by the currents there will be a residual current not neutralized 
by any other; and therefore the result will be the same as that of a single 
unit current round the boundary of all the currents. 

From this it appears that the external attractions of a shell uniformly 
magnetized perpendicular to its surface are the same as those due to a current 
round its edge, for each of the elementary currents in the former case has 
the same effect as an element of the magnetic shell. 

If we examine the Unes of magnetic force produced by a closed current, 
we shall find that they form closed curves passing round the current and 
embracing it, and that the total intensity of the magnetizing force all along 
the closed line of force depends on the quantity of the electric current only. 



ON FARADAY 3 LINES OF FORCE. 185 

The number of unit lines* of magnetic force due to a closed current depends 
on the form as well as the quantity of the current, but the number of unit 
cells t in each complete line of force is measured simply by the number of unit 
currents which embrace it. The unit cells in this case are portions of space in 
which unit of magnetic quantity is produced by unity of magnetizing force. 
The length of a cell is therefore inversely as the intensity of the magnetizing 
force, and its section inversely as the quantity of magnetic induction at that 
point. 

The whole number of cells due to a given current is therefore proportional 
to the strength of the current multiplied by the number of lines of force 
which pass through it. If by any change of the form of the conductors the 
number of cells can be increased, there will be a force tending to produce that 
change, so that there is always a force urging a conductor transverse to the 
lines of magnetic force, so as to cause more lines of force to pass throuo-h the 
closed circuit of which the conductor forms a part. 

The number of cells due to two given currents is got by multiplying 
the number of lines of inductive magnetic action which pass through each by 
the quantity of the currents respectively. Now by (9) the number of lines 
which pass through the first current is the sum of its own lines and those 
of the second current which would pass through the first if the second current 
alone were in action. Hence the whole number of cells will be increased by 
any motion which causes more lines of force to pass through either circuit, 
and therefore the resultant force will tend to produce such a motion, and the 
work done by this force during the motion will be measured by the number 
of new cells produced. All the actions of closed conductors on each other may 
be deduced from this principle. 

On Electric Currents prodiiced by Induction. 

Faraday has shewn| that when a conductor moves transversely to the lines 
of magnetic force, an electro-motive force arises in the conductor, tending to 
produce a current in it. If the conductor is closed, there is a continuous 
current, if open, tension is the result. If a closed conductor move transversely 
to the lines of magnetic induction, then, if the number of lines which pass 

♦ Hxp. Rea. (3122). See Art. (6) of this paper. t Art. (13). 

X Exp. lies. (3077), &c. 

VOL. I. 24 



186 ON Faraday's lines of force. 

through it does not change during the motion, the electro-motive forces in the 
circuit will be in equilibrium, and there will be no current. Hence the electro- 
motive forces depend on the number of lines which are cut by the conductor 
during the motion. If the motion be such that a greater number of lines pass 
through the circuit formed by the conductor after than before the motion, 
then the electro-motive force will be measured by the increase of the number 
of lines, and will generate a current the reverse of that which would have 
produced the additional Hnes. When the number of lines of inductive magnetic 
action through the circuit is increased, the induced current will tend to diminish 
the number of lines, and when the number is diminished the induced current 
will tend to increase them. 

That this is the true expression for the law of induced currents is shewn 
from the fact that, in whatever way the number of lines of magnetic induction 
passing through the circuit be increased, the electro-motive effect is the same, 
whether the increase take place by the motion of the conductor itself, or of other 
conductors, or of magnets, or by the change of intensity of other currents, or 
by the magnetization or demagnetization of neighbouring magnetic bodies, or 
lastly by the change of intensity of the current itself. 

In all these cases the electro-motive force depends on the change in the 
number of lines of inductive magnetic action which pass through the circuit*. 

* The electro-magnetic forces, which tend to produce motion of the material conductor, must be 
carefully distinguished from the electro-motive forces, which tend to produce electric currents. 

Let an electric current be passed through a mass of metal of any form. The distribution of 
the currents within the metal will be determined by the laws of conduction. Now let a constant 
electric cuiTent be passed through another conductor near the first. If the two currents are in the 
same direction the two conductors will be attracted towards each other, and would come nearer if 
not held in their positions. But though the material conductors are attracted, the currents (which 
are free to choose any course within the metal) will not alter their original distribution, or incline 
towards each other. For, since no change takes place in the system, there will be no electro-motive 
forces to modify the original distribution of currents. 

In this case we have electro-magnetic forces acting on the material conductor, without any 
electi"o-motive forces tending to modify the current which it can-ies. 

Let us take as another example the case of a linear conductor, not forming a closed circuit, 
and let it be made to traverse the lines of magnetic force, either by its own motion, or by changes 
in the magnetic field. An electro-motive force wiU act in the direction of the conductor, and, as it 
cannot produce a current, because there is no circuit, it will produce electric tension at the extremi- 
ties. There will be no electro-magnetic attraction on the material conductor, for this attraction 
depends on the existence of the cun-ent within it, and this is prevented by the circuit not being closed. 

Here then we have the opposite case of an electro-motive force acting on the electricity in the 
conductor, but no attraction on its material particles. 



ON FARADAY 8 LINES OF FORCE. 187 

It is natural to suppose that a force of this kind, which depends on a 
change in the number of lines, is due to a change of state which is measured 
by the number of these lines. A closed conductor in a magnetic field may- 
be supposed to be in a certain state arising from the magnetic action. 
As long as this state remains unchanged no effect takes place, but, when the 
state changes, electro-motive forces arise, depending as to their intensity and 
direction on this change of state. I cannot do better here than quote a 
passage from the first series of Faraday's Experimental Researches, Art. (60). 

"While the wire is subject to either volta-electric or magno-electric 
induction it appears to be in a peculiar state, for it resists the formation of 
an electrical current in it ; whereas, if in its common condition, such a current 
would be produced; and when left uninfluenced it has the power of originating a 
current, a power which the wire does not possess under ordinary circumstances. 
This electrical condition of matter has not hitherto been recognised, but it 
probably exerts a very important influence in many if not most of the phe- 
nomena produced by currents of electricity. For reasons which will immediately 
appear (7) I have, after advising with several learned friends, ventured to 
designate it as the electro-tonic state." Finding that all the phenomena could 
be otherwise explained without reference to the electro-tonic state, Faraday in 
his second series rejected it as not necessary ; but in his recent researches ■'"' 
he seems still to think that there may be some physical truth in his 
conjecture about this new state of bodies. 

The conjecture of a philosopher so familiar with nature may sometimes be 
more pregnant with truth than the best established experimental law disco- 
vered by empirical inquirers, and though not bound to admit it as a physical 
truth, we may accept it as a new idea by which our mathematical conceptions 
may be rendered clearer. 

In this outline of Faraday's electrical theories, as they appear from a 
mathematical point of view, I can do no more than simply state the mathe- 
matical methods by which I believe that electrical phenomena can be best 
comprehended and reduced to calculation, and my aim has been to present the 
mathematical ideas to the mind in an embodied form, as systems of lines or 
surfaces, and not as mere symbols, which neither convey the same ideas, nor 
readily adapt themselves to the phenomena to be explained. The idea of the 
electro-tonic state, however, has not yet presented itself to my mind in such a 

* (3172) (3269). 

24—2 



188 ON Faraday's lines of force. 

form that its nature and properties may be clearly explained witliout reference 
to mere symbols, and therefore I propose in the following investigation to use 
symbols freely, and to take for granted the ordinary mathematical operations. 
By a careful study of the laws of elastic solids and of the motions of viscous 
fluids, I hope to discover a method of forming a mechanical conception of this 
electro- tonic state adapted to general reasoning*. 



Part II. 

On Faraday's " Electro^tonic State" 

When a conductor moves in the neighbourhood of a current of electricity, 
or of a magnet, or when a current or magnet near the conductor is moved, or 
altered in intensity, then a force acts on the conductor and produces electric 
tension, or a continuous current, according as the circuit is open or closed. This 
current is produced only by changes of the electric or magnetic phenomena sur- 
rounding the conductor, and as long as these are constant there is no observed 
effect on the conductor. Still the conductor is in different states when near- a 
current or magnet, and when away from its influence, since the removal or 
destruction of the current or magnet occasions a current, which would not have 
existed if the magnet or current had not been previously in action. 

Considerations of this kind led Professor Faraday to connect with his 

discovery of the induction of electric currents the conception of a state into 
which all bodies are thrown by the presence of magnets and currents. This 
state does not manifest itself by any known phenomena as long as it is undis- 
turbed, but any change in this state is indicated by a current or tendency 
towards a current. To this state he gave the name of the " Electro-tonic 
State," and although he afterwards succeeded in explaining the phenomena 
which suggested it by means of less hypothetical conceptions, he has on several 
occasions hinted at the probability that some phenomena might be discovered 
which would render the electro-tonic state an object of legitimate induction. 
These speculations, into which Faraday had been led by the study of laws 
which he has well established, and which he abandoned only for want of experi- 

* See Pro£ W. Thomson On a Mechanical Representation of Electric, Magnetic and Galvanic 
Forces. Camvb. and Dub. Math. Jour. Jan. 1847. 



ON f^vraday's lines of forcr 189 

mental data for the direct proof of the unknown state, have not, I think, been 
made the subject of mathematical investigation. Perhaps it may be thought 
that the quantitative determinations of the various phenomena are not suffi- 
ciently rigorous to be made the basis of a mathematical theory ; Faraday, 
however, has not contented himself with simply stating the numerical results of 
his experiments and leaving the law to be discovered by calculation. Where 
he has perceived a law he has at once stated it, in terms as unambiguous as 
those of pure mathematics ; and if the mathematician, receiving this as a physical 
truth, deduces from it other laws capable of being tested by experiment, he 
has merely assisted the physicist in arranging his own ideas, which is con- 
fessedly a necessary step in scientific induction. 

In the following investigation, therefore, the laws established by Faraday 
will be assumed aa true, and it will be shewn that by following out his 
speculations other and more general laws can be deduced from them. If it 
should then appear that these laws, originally devised to include one set of 
phenomena, may be generalized so as to extend to phenomena of a different 
class, these mathematical connexions may suggest to physicists the means of 
establishing physical connexions; and thus mere speculation may be turned to 
account in experimental science. 

On Quantity and Intensity as Properties of Electric Currents. 

It is found that certain effects of an electric current are equal at what- 
ever part of the circuit they are estimated. The quantities of water or of 
any other electrolyte decomposed at two different sections of the same circuit, 
are always found to be equal or equivalent, however different the material and 
form of the circuit may be at the two sections. The magnetic effect of a 
conducting wire is also found to be independent of the form or material of 
the wire in the same circuit. There is therefore an electrical effect which is 
equal at every section of the circuit. If we conceive of the conductor as the 
channel along which a fluid is constrained to move, then the quantity of fluid 
transmitted by each section will be the same, and we may define the quantity 
of an electric current to be the quantity of electricity which passes across a 
complete section of the current in unit of time. We may for the present 
measure quantity of electricity by the quantity of water which it would decom- 
pose in unit of time. 



190 ON FABADAYS LINES OF FORCE. 

In order to express mathematically the electrical currents in any conductor, 
we must have a definition, not only of the entire flow across a complete section, 
but also of the flow at a given point in a given direction. 

Def. The quantity of a current at a given point and in a given direction 
is measured, when uniform, by the quantity of electricity which flows across 
unit of area taken at that point perpendicular to the given direction, and when 
variable by the quantity which would flow across this area, supposing the flow 
uniformly the same as at the given point. 

In the following investigation, the quantity of electric current at the point 
(xyz) estimated in the directions of the axes x, y, z respectively will be denoted 
by Oj, 5j, C3. 

The quantity of electricity which flows in unit of time through the ele- 
mentary area dS 

= dS (la^ + ?nZ)2 + nc^), 
where I, m, n are the direction-cosines of the normal to dS. 

This flow of electricity at any point of a conductor is due to the electro- 
motive forces which act at that point. These may be either external or internal. 

External electro- motive forces arise either from the relative motion of currents 
and magnets, or from changes in their intensity, or from other causes acting 
at a distance. 

Internal electro-motive forces arise principally from diSerence of electric 
tension at points of the conductor in the immediate neighbourhood of the point 
in question. The other causes are variations of chemical composition or of tem- 
perature in contiguous parts of the conductor. 

Let Pi represent the electric tension at any point, and X^, F,, Z, the sums 
of the parts of all the electro-motive forces arising from other causes resolved 
parallel to the co-ordinate axes, then if Og, ySj, y^ be the efiective electro-motive 
forces 

"^-^^'dx 



dp, 
^'-^'"dy 

dp, 

y^^^'^-d^ 



(A). 



ON Faraday's lines of force. 191 

Now the quantity of the current depends on the electro-motive force and 
on the resistance of the medium. If the resistance of the medium be uniform 
in all directions and equal to k^, 

a^ = Jc,a„ ^, = kK y2 = Kc2 (B), 

but if the resistance be different in different directions, the law will be more 
complicated. 

These quantities Oj, /3j, y., may be considered as representing the intensity 
of the electric action in the directions of x, y, z. 

The intensity measured along an element da of a curve is given by 

€ = Za + mji + ny, 
where Z, m, n are the direction-cosines of the tangent. 

The integral JecZcr taken with respect to a given portion of a curve line, 
represents the total intensity along that line. If the curve is a closed one, it 
represents the total intensity of the electro-motive force in the closed curve. 

Substituting the values of a, /8, y from equations (A) 

l^da- = l{Xdx + Ydy + Zdz) -p + a 

If therefore {Xdx+ Ydy + Zdz) is a complete differential, the value of Jedo- for 
a closed curve will vanish, and in all closed curves 

leda- = l{Xdx+Ydy + Zdz), 

the integration being effected along the curve, so that in a closed curve the 
total intensity of the effective electro- motive force is equal to the total intensity 
of the impressed electro-motive force. 

The total quantity of conduction through any surface is expressed by 

\edS, 
where 

e = la + mh + nc, 

I, m, n being the direction- cosines of the normal, 

. •. \edS = l\adydz + ^bdzdx + \\cdxdy, 

the integrations being effected over the given surface. AVhen the surface is a 
closed one, then we may find by integration by parts 



w.=///(:- 



7a dh dc\ , , , 



192 ON FARADAY S LINES OF FORCE. 

If we make 

da dh ^ d.c /^v 

Tx + dy+di^^^P (^)' 

\edS= iirlWpdxdydz, 

where the integration on the right side of the equation is effected over every 
part of space within the surface. In a large class of phenomena, including all 
cases of uniform currents, the quantity p disappears. 



Magnetic Quantity and Intensity. 

From his study of the lines of magnetic force, Faraday has been led to 
the conclusion that in the tubular surface ■''' formed by a system of such lines, 
the quantity of magnetic induction across any section of the tube is constant, 
and that the alteration of the character of these lines in passing from one 
substance to another, is to be explained by a difference of inductive capacity 
in the two substances, which is analogous to conductive power in the theory 
of electric currents. 

In the following investigation we shall have occasion to treat of magnetic 
quantity and intensity in connection with electric. In such cases the magnetic 
symbols wiU be distinguished by the sufiix 1, and the electric by the suffix 2. 
The equations connecting a, h, c, h, a, /8, y, p, and p, are the same in form as 
those which we have just given, a, 6, c are the symbols of magnetic induction 
with respect to quantity ; k denotes the resistance to magnetic induction, and 
may be different in different directions ; a, /8, y, are the effective magnetiang 
forces, connected with a, h, c, by equations (B) ; p is the magnetic tension or 
potential which will be afterwards explained ; p denotes the density of real 
magnetic matter and is connected with a, h, c by equations (C). As all the 
details of magnetic calculations will be more intelligible after the exposition of the 
connexion of magnetism with electricity, it will be sufficient here to say that 
all the definitions of total quantity, with respect to a surface, the total intensity 
to a curve, apply to the case of magnetism as well as to that of electricity. 

* Exp. Res. 3271, definition of " Sphondyloid." 



ON Faraday's lines of force. 193 



Electro-magnetism. 



Ampere has proved the following laws of the attractions and repulsions of 
electric currents : 

I. Equal and opposite currents generate equal and opposite forces. 

II. A crooked current is equivalent to a straight one, provided the two 
currents nearly coincide throughout their whole length. 

IIL Equal currents traversing similar and similarly situated closed curves 
act with equal forces, whatever be the linear dimensions of the circuits. 

IV. A closed current exerts no force tending to turn a circular conductor 
about its centre. 

It is to be observed, that the currents with which Ampere worked were constant 
and therefore re-entering. All his results are therefore deduced from experiments 
on closed currents, and his expressions for the mutual action of the elements 
of a current involve the assumption that this action is exerted in the direction 
of the line joining those elements. This assumption is no doubt warranted by the 
universal consent of men of science in treating of attractive forces considered 
as due to the mutual action of particles ; but at present we are proceeding 
on a different principle, and searching for the explanation of the phenomena, 
not in the currents alone, but also in the surrounding medium. 

The first and second laws shew that currents are to be combined like 
velocities or forces. 

The third law is the expression of a property of all attractions which may 
be conceived of as depending on the inverse square of the distance from a fixed 
system of points ; and the fourth shews that the electro-magnetic forces may 
always be reduced to the attractions and repulsions of imaginary matter properly 
distributed. 

In fact, the action of a very small electric circuit on a point in its neigh- 
bourhood is identical with that of a small magnetic element on a point outside 
it. If we divide any given portion of a surface into elementary areas, and 
cause equal currents to flow in the same direction round all these Httle areas, 
the effect on a point not in the surface will be the same as that of a shell 
coinciding with the surface, and uniformly magnetized normal to its surface. 
But by the first law all the currents forming the little circuits will destroy 

VOL. L 25 



194 ON FARADAY S LINES OF FORCE. 

one another, and leave a single current running round the bounding line. So 
that the magnetic effect of a uniformly magnetized shell is equivalent to that 
of an electric current round the edge of the shell. If the direction of the current 
coincide with that of the apparent motion of the sun, then the direction of 
magnetization of the imaginary shell will be the same as that of the real mag- 
netization of the earth*. 

The total intensity of magnetizing force in a closed curve passing through 
and embracing the closed current is constant, and may therefore be made a 
measure of the quantity of the current. As this intensity is independent of the 
form of the closed curve and depends only on the quantity of the current which 
passes through it, we may consider the elementary case of the current which 
Hows through the elementary area dydz. 

Let the axis of x point towards the west, z towards the south, and y 
upwards. Let x, y, z be the coordinates of a point in the middle of the area 
dydz, then the total intensity measured round the four sides of tlie element is 



(A*Si)* 


('■* 


t' 1') *. 
dy 2j 


{*- 


■ff)^^. 


('- 


■tf)<'^- 


[dz 


-©''^*- 



Total intensity = 

The quantity of electricity conducted through the elementary area dydz is 
adydz, and therefore if we define the measure of an electric current to be the 
total intensity of magnetizing force in a closed curve embracing it, we shall have 

^^^dl,_dy, 
' dz dy ' 



h,. 



dy^ dai 
dx dz 



_da,_d£, 
' dy dx 



-■ See Experimental Researches (3265) for the relations between the electrical and magnetic circuit, 
considered as mutiudly embracing curves. 



ON Faraday's lines of force. 195 

These equations enable us to deduce the distribution of the currents of 
electricity whenever we know the values of a, y3, y, the magnetic intensities. 
If a, /3, y be exact differentials of a function of x, y, z with respect to x, y 
and 2 respectively, then the values of a,, h^, c, disappear; and we know that the 
magnetism is not produced by electric currents in that part of the field which 
we are investigating. It is due either to the presence of permanent magnetism 
within the field, or to magnetising forces due to external causes. 

We may observe that the above equations give by differentiation 

^ + ^'4.^^ = 

dx dy dz * 

which is the equation of continuity for closed currents. Our investigations are 
therefore for the present limited to closed currents ; and we know little of the 
magnetic effects of any currents which are not closed. 

Before entering on the calculation of these electric and magnetic states it 
may be advantageous to state certain general theorems, the truth of which may 
be established analytically. 

Theorem I. 

The equation 

d'V d^V d'V ^ ^ 

d^-^W'^^'^ ^^^ ' 

(where V and p are functions of x, y, z never infinite, and vanishing for all points 
at an infinite distance), can be satisfied by one, and only one, value of V. See 
Art. (17) above. 



Theorem II. 

The value of V which will satisfy the above conditions is found by inte- 
grating the expression 

pdxdydz 



///, 



where the limits of x, 3/, 2 are such as to include every point of space where /> 
is finite. 

25—2 



196 ON Faraday's lines of force. 

The proofs of these theorems may be found in any work on attractions or 

electricity, and in particular in Green's Essay on the Application of Mathematics 

to Electricity. See Arts. 18, 19 of this paper. See also Gauss, on Attractions^ 
translated in Taylor's Scientijtc Memoirs. 



Theorem III. 
Let U and V be two functions of x, y, z, then 

d'U d'U d'-U\ J., , , 

where the integrations are supposed to extend over all the space in which U 
and V have values differing from 0. — (Green, p. 10.) 

This theorem shews that if there be two attracting systems the actions 
between them are equal and opposite. And by making U= V we find that 
the potential of a system on itself is proportional to the integral of the square 
of the resultant attraction through all space ; a result deducible from Art. (30), 
since the volume of each cell is inversely as the square of the velocity (Arts. 
12, 13), and therefore the number of cells in a given space is directly as the 
square of the velocity. 

Theorem IV. 

Let a, /8, y, p be quantities finite through a certain space and vanishing 
in the space beyond, and let k be given for all parts of space as a continuous 
or discontinuous function of x, y, z, then the equation in p 

has one, and only one solution, in which p is always finite and vanishes at 
an infinite distance. 

The proof of this theorem, by Prof W. Thomson, may be found in the 
Cambridge and Dublin Mathematical Journal, Jan. 1848. 



ON FARADAY S LINES OF FORCE, 197 

If a, /3, y be the electro-motive forces, p the electric tension, and Ic the 
coefficient of resistance, tlien the above equation is identical with the equation 
of continuity 

da^ ,dh,dc, 

ax dy dz r ' 

and the theorem shews that when the electro-motive forces and the rate of 
production of electricity at every part of space are given, the value of the 
electric tension is determinate. 

Since the mathematical laws of magnetism are identical with those of elec- 
tricity, as far as we now consider them, we may regard a, /8, y as magnetizing 
forces, p as magnetic tension, and p as real magnetic density, k being the 
coefficient of resistance to magnetic induction. 

The proof of this theorem rests on the determination of the minimum value 

where V is got from the equation 

d'V d'V d'V , 

and p has to be determined. 

The meaning of this integral in electrical language may be thus brought 
out. If the presence of the media in which k has various values did not 
affect the distribution of forces, then the '^quantity" resolved in x would be 

simply -7— and the intensity k -^ . But the actual quantity and intensity are 

J- (a — j-j and a— ^, and the parts due to the distribution of media alone 
are therefore 



1 / dp\ dV , dp , dV 
T {°'-~ji — 7- and a — ~ — k -i- . 
fc \ ax) dx dx dx 



Now the product of these represents the work done on account of this 
distribution of media, the distribution of sources being determined, and taking 
in the terms in y and z we get the expression Q for the total work done 



198 ON Faraday's lines or force. 

by that part of the whole effect at any point which is due to the distribution 
of conducting media, and not directly to the presence of the sources. 

This quantity Q is rendered a minimum by one and only one value of p, 
namely, that which satisfies the original equation. 

Theorem V. 
If a, h, c be three functions of x, y, % satisfying the equation 

da db ^ _r. 
dx dy dz~ ' 

it is always possible to find three functions a, /3, y which shall satisfy the equa- 
tions 

dz dy ' 



i- 


da 


-h, 


da 


dfi 

-Tx'- 


= c. 



Let A = Icdy, where the integration is to be performed upon c considered 
as a function of y, treating x and z as constants. Let B='\adz, C^\hdx, 
A' = \hdz, R = \cdx, C' = \ady, integrated in the same way. 

Then 

will satisfy the given equations ; for 

d§^_dy^fda^^^fdc^^__fdb^^_^fda , 
dz dy J dy J dz Jdy J dy ^' 

and 0=\--j-dx+\-f- dx+ l-j- dx; 

d3 dy (da , (da , (da , 

= a. 



ON Faraday's lines of force. 199 

In the same way it may be shewn that the values of a, ^, y satisfy 
the other given equations. The function i/; may be considered at present as 
perfectly indeterminate. 

The method here given is taken from Prof. W. Thomson's memoir on 
Magnetism {Phil Trans. 1851, p. 283). 

As we cannot perform the required integrations when a, h, c are discon- 
tinuous functions of x, y, z, the following method, which is perfectly general 
though more compUcated, may indicate more clearly the truth of the proposition. 

Let A, B, C be determined from the equations 
d'A d'A d'A 

^ + ^^ + £^ + 6 = 
dor dy^ dz' ' 

d'Cd'Ccr-c^ ^ 

by the methods of Theorems I. and II., so that A, B, C are never infinite, 
and vanish when x, y, or z is infinite. 
Also let 



then 



a = 


dB 

-dz' 


dC 
-dy- 


d^ 
^dx' 






0- 


dC 
~ dx' 


dA 

"dz' 


dy 






7- 


dA 
-dy- 


dB 
~dx' 


drP 

^Tz' 






^ 


dB 

-dy' 


dC\ fd'A d'A 
- dz) W "^ clf "^ 


d"-A 
dz\ 



d^/dA dB dC\,^ 
dx\dx dy dzj 



dx\dx dy 

If we find similar equations in y and z, and differentiate the first by x, 
the second by y, and the third by z, remembering the equation between 
a, b, c, we shall have 

/c?^ d^ dr\fdA dB cZC\ 

\dxr dif' dz^]\dx dy dz 



200 ON Faraday's lines of force. 

and since A, B, C are always finite and vanish at an ir finite distance, the 
only solution of this equation is 

dA dB dC^^ 
dx dy dz * 

and we have finally 

d§ _dY_ 
dz dy~ ' 

with two similar equations, shewing that a, /9, y have been rightly determined. 

The function i/» is to be determined from the condition 

dx^ dy^ dz~ [dx" '^dy'^ dz') ^ ' 
if the left-hand side of this equation be always zero, xp must be zero also. 



Theorem YI, 

Let a, h, c he any three functions of x, y, z, it is possible to find three 
functions a, /8, y and a fourth V, so that 

dx dy dz ' 

and =^_^ ^ 

dz dy dx ' 

,_dy^dadV 
dx dz dy ' 

dy dx dz 

Let 

da dh dc 

di + Ty + dz^-^^'P' 

and let V be found from the equation 

d^V d'V d'V 



ON fabaday's lines of force. 201 

then 





a'^a- 


dV 
dx' 




h' = h- 


dV 




c=c- 


dV 


da' 
-dbc 


dh' 


dc ^ 
dz 



satisfy the condition 



and therefore we can find three tunctions A, B, C, and from these a, ^, y, so as 
to satisfy the given equations. 

Theorem VIL 

The integral throughout infinity 

Q = jjj (a,a, + hfi, + c^y,) dxdydz, 
where a}>fi^, a^{y^ are any functions whatsoever, is capable of transformation into 

Q=+ lll{^n>P^ - (^o«2 + A^2 + roC,)} dxdydz, 
in which the quantities are found from the equations 

dcL dh, d€< 
^■'dy^fz^'^P^-''^ 

ojSoyo^ axe determined from ap^c^ by the last theorem, so that 

^ dz dy dx ' 
a}>/:^ are found from cgSiyi by the equations 

and p is found from the equation 

d'p d'pd'p^^ , . 

vol. l 26 



202 ON Faraday's lines of force. 

For, if we put a, in the form 

dz dy dx ' 

and treat h^ and c, similarly, then we have by integration by parts through 
infinity, remembering that all the functions vanish at the limits, 

or <? = + ///{(47r V) - (aA + A&. + y.c,)] dxdydz, 
and by Theorem III. 

Ill Vp dxdydz = lUppdxdydz, 
so that finally 

Q = lll{^7rpp - (a„a, + A^2 + y«cj} dxdydz. 

If afi^c^ represent the components of magnetic quantity, and a^iyi those 
of magnetic intensity, then p will represent the real magnetic density, and p 
the magnetic potential or tension. aJ)iCi will be the components of quantity 
of electric currents, and a^^.y^ will be three functions deduced from afi^c^, 
which will be found to be the mathematical expression for Faraday's Electro- 
tonic state. 

Let us now consider the bearing of these analytical theorems on the 
theory of magnetism. Whenever we deal with quantities relating to magnetism, 
we shall distinguish them by the suffix d). Thus aj^iC, are the components 
resolved in the directions of x, y, z of the quantity of magnetic induction acting 
through a given point, and aJS^yi are the resolved intensities of magnetization 
at the same point, or, what is the same thing, the components of the force 
which would be exerted on a unit south pole of a magnet placed at that 
point without disturbing the distribution of magnetism. 

The electric currents are found from the magnetic intensities by the equations 

djB, dy, , 
dz dy 

When there are no electric currents, then 

a^dx + P^dy -f y^dz = dp, , 



ON Faraday's lines of force. 203 

a perfect differential of a function of x, y, z. On the principle of analogy we 
may call jo, the magnetic tension. 

The forces which act on a mass m of south magnetism at any point are 

in the direction of the axes, and therefore the whob work done during any 
displacement of a magnetic system is equal to the decrement of the integral 

Q = ll\p,p4xdydz 
throughout the system. 

Let us now call Q the total potential of the system on itself. The increase 
or decrease of Q will measure the work lost or gained by any displacement 
of any part of the system, and will therefore enable us to determine the 
forces acting on that part of the system. 

By Theorem III. Q may be put under the form 



Q = + ^ j I (ctio, + hSi + c,y,) dxdydz 



in which a^iji are the differential coefficients of p^ with respect to x, y, z 
respectively. 

If we now assume that this expression for Q is true whatever be the 
values of Oj, )8„ yi, we pass from the consideration of the magnetism of permanent 
magnets to that of the magnetic effects of electric currents, and we have then 
by Theorem VII. 

So that in the case of electric currents, the components of the currents have 
to be multiplied by the functions a„, ySj, yo respectively, and the summations of 
all such products throughout the system gives us the part of Q due to those 
currents. 

We have now obtained in the functions a,,, Aj yo the means of avoiding 
the consideration of the quantity of magnetic induction which passes through 
the circuit. Instead of this artificial method we have the natural one of con- 
sidering the current with reference to quantities existing in the same space 
with the current itself. To these I give the name of Electro-tonic functions, or 
components of the Electro-tonic intensity. 

2G— 2 



204 ON Faraday's lines of force. 

Let us now consider the conditions of the conduction of the electric 
currents within the medium during changes in the electro-tonic state. The 
method which we shall adopt is an appHcation of that given by Helmholtz in 
his memoir on the Conservation of Force*. 

Let there be some external source of electric currents which would generate 
in the conducting mass currents whose quantity is measured by a^, h^, c, and 
their intensity by cu,, /Sa, y^. 

Then the amount of work due to this cause in the time dt is 
dt lll{a^(h + hS^ + c^y^ dxdydz 
in the form of resistance overcome, and 

^ ^ J j J (^2^0 4- 6 A + c,yo) dxdydz 

in the form of work done mechanically by the electro-magnetic action of these 
currents. If there be no external cause producing currents, then the quantity 
representing the whole work done by the external cause must vanish, and we 
have 

dt \\ \(a,a^ + hS, + c.y,) dxdydz + 4^ ^ I I I («**o + ^So + c^Jo) dxdydz, 

where the integrals are taken through any arbitrary space. We must therefore 
have 

for every point of space ; and it must be remembered that the variation of 
Q is supposed due to variations of a^, ySo, y^, and not of a^, \, c^. We must 
therefore treat a^, 63, c^ as constants, and the equation becomes 

In order that this equation may be independent of the values of a^, b^, Cj, 
each of these coefficients must = ; and therefore we have the following 
expressions for the electro-motive forces due to the action of magnets and 
currents at a distance in terms of the electro-tonic functions, 

°^~ ATrdt' ^^~ Andt' '^'~ An dt ' 

* Translated in Taylor's N'ew Scientific Memoirs, Part 11. 



ON Faraday's lines of force. 205 

It appears from experiment that the expression -jj refers to the change 

of electro-tonic state of a given particle of the conductor, whether due to 
change in the electro-tonic functions themselves or to the motion of the particle. 

If Oo be expressed as a function of x, y, z and t, and \£ x, y, z be the 
co-ordinates of a moving particle, then the electro-motive force measured in the 
direction of a; is 



_ _ Jl (^' dx da^dy da,dz doA 
°^~ 477 \dx dt dy dt dz dt dtj 



The expressions for the electro-motive forces in y and z are similar. The 
distribution of currents due to these forces depends on the form and arrange- 
ment of the conducting media and on the resultant electric tension at any 
point. 

The discussion of these functions would involve us in mathematical formulae, 
of which this paper is already too full. It is only on account of their physical 
importance as the mathematical expression of one of Faraday's conjectures that I 
have been induced to exhibit them at all in their present form. By a more 
patient consideration of their relations, and with the help of those who are 
engaged in physical inquiries both in this subject and in others not obviously 
connected with it, I hope to exhibit the theory of the electro-tonic state in a 
form in which all its relations may be distinctly conceived without reference to 
analytical calculations. 



Summary of the Theory of the Electro-tonic State. 

We may conceive of the electro-tonic state at any point of space as a 
quantity determinate in magnitude and direction, and we may represent the 
electro-tonic condition of a portion of space by any mechanical system which 
has at every point some quantity, which may be a velocity, a displacement, or 
a force, whose direction and magnitude correspond to those of the supposed 
electro-tonic state. This representation involves no physical theory, it is only 
a kind of artificial notation. In analytical investigations we make use of the 
three components of the electro-tonic state, and call them electro-tonic functions. 
We take the resolved part of the electro-tonic intensity at every point of a 



206 ON Faraday's lines of force. 

closed curve, and find by integration what we may caU the entire electro-tonic 
intensity round the curve. 

Prop. I. If on any surface a closed curve be drawn, and if the surface 
within it he divided into small areas, then the entire intensity round the closed 
curve is equal to the sum of the intensities round each of the small areas, all 
estimated in the same direction. 

For, in going round the small areas, every boundary line between two of 
them is passed along twice in opposite directions, and the intensity gained in 
the one case is lost in the other. Every eflfect of passing along the interior 
divisions is therefore neutraUzed, and the whole efiect is that due to the 
exterior closed curve. 

Law I. The entire dectro-tonic intensity round the boundary of an element of 
surface measures the quantity of magnetic induction which passes through that 
surface, or, in other words, the number of lines of magnetic force which pass 
through that surface. 

By Prop. I. it appears that what is true of elementary surfaces is true also 
of surfaces of finite magnitude, and therefore any two surfaces which are 
bounded by the same closed curve will have the same quantity of magnetic 
induction through them. 

Law II. The magnetic intensity at any point is connected with the quantity 
of magnetic induction by a set of linear equations, called the equations of con- 
duction*. 

Law III. The entire magnetic intensity round the boundary of any surface 
measures the quantity of electric current which passes through that surface. 

Law IV. The quantity and intensity of electric currents are connected by a 
system of equations of conduction. 

By these four laws the magnetic and electric quantity and intensity may be 
deduced from the values of the electro-tonic functions. I have not discussed 
the values of the units, as that will be better done with reference to actual 
experiments. We come next to the attraction of conductors of currents, and to 
the induction of currents within conductors. 

* See Art. (28). 



ON FARADAY S LINES OF FORCE. 207 

Law v. The total electro-magnetic potential of a closed current is measxired 
by the product of the quantity of the current multiplied by the entire electro-tonic 
intensity estimated in t/ie same direction round the circuit. 

Any displacement of the conductors which would cause an increase in the 
potential will be assisted by a force measured by the rate of increase of the 
potential, so that the mechanical work done during the displacement will be 
measured by the increase of potential. 

Although in certain cases a displacement in direction or alteration of inten- 
sity of the current might increase the potential, such an alteration would not 
itself produce work, and there will be no tendency towards this displacement, 
for alterations in the current are due to electro-motive force, not to electro- 
magnetic attractions, which can only act on the conductor. 

Law VI. The electro-motive force on any element of a conductor is measured 
by the instantaneous rate of change of the electro-tonic intensity on that element, 
whether in magnitude or direction. 

The electro-motive force in a closed conductor is measured by the rate of 
change of the entire electro-tonic intensity round the circuit referred to unit 
of time. It is independent of the nature of the conductor, though the current 
produced varies inversely as the resistance ; and it is the same in whatever 
way the change of electro-tonic intensity has been produced, whether by motion 
of the conductor or by alterations in the external circumstances. 

In these six laws I have endeavoured to express the idea which I believe to 
be the mathematical foundation of the modes of thought indicated in the Ex- 
perimental Researches. I do not think that it contains even the shadow of a 
true physical theory; in fact, its chief merit as a temporary instrument of 
research is that it does not, even in appearance, account for anything. 

There exists however a professedly physical theory of electro-dynamics, which 
is so elegant, so mathematical, and so entirely different from anything in this 
paper, that I must state its axioms, at the risk of repeating what ought to 
be well known. It is contained in M. W. Weber's Electro-dynamic Measure- 
ments, and may be found in the Transactions of the Leibnitz Society, and of the 
Royal Society of Sciences of Saxony*. The assumptions are, 

* When this was written, I was not aware that part of M. Weber's Memoir is translated in 
Taylor's Scientific Memoirs, VoL v. Art. xiv. The value of his researches, both experimental and 
theoretical, renders the study of his theory necessary to every electrician. 



208 ON Faraday's lines of force. 

(1) That two particles of electricity when in motion do not repel each other 
with the same force as when at rest, but that the force is altered by a quantity 
depending on the relative motion of the two particles, so that the expression for 
the repulsion at distance r is 



eeV, dr 






(2) That when electricity is moving in a conductor, the velocity of the 
positive fluid relatively to the matter of the conductor is equal and opposite to 
that of the negative fluid. 

(3) The total action of one conducting element on another is the resultant 
of the mutual actions of the masses of electricity of both kinds which are 
in each. 

(4) The electro-motive force at any point is the difference of the forces 
acting on the positive and negative fluids. 

From these axioms are deducible Ampere's laws of the attraction of 
conductors, and those of Neumann and others, for the induction of currents. 
Here then is a really physical theory, satisfying the required conditions better 
perhaps than any yet invented, and put forth by a philosopher whose experi- 
mental researches form an ample foundation for his mathematical investigations. 
What is the use then of imagining an electro-tonic state of which we have 
no distinctly physical conception, instead of a formula of attraction which we 
can readily understand ? I would answer, that it is a good thing to have 
two ways of looking at a subject, and to admit that there are two ways of 
looking at it. Besides, I do not think that we have any right at present to 
understand the action of electricity, and I hold that the chief merit of a 
temporary theory is, that it shall guide experiment, without impeding the 
progress of the true theory when it appears. There are also objections to 
making any ultimate forces in nature depend on the velocity of the bodies 
between which they act. If the forces in nature are to be reduced to forces 
acting between particles, the principle of the Conservation of Force requires 
that these forces should be in the line joining the particles and functions of 
the distance only. The experiments of M. Weber on the reverse polarity of 
diaraagnetics, which have been recently repeated by Professor Tyndall, establish 
a fact which is equally a consequence of M. Weber's theory of electricity and 
of the theory of lines of fcH-ce. 



ON FARADAY S LINES OF FORCE. 209 

With respect to the history of the present theory, I may state that the 
recognition of certain mathematical functions as expressing the "electro-tonic 
state " of Faraday, and the use of them in determining electro-dynamic 
potentials and electro-motive forces is, as far as I am aware, original ; but the 
distinct conception of the possibility of the mathematical expressions arose in 
my mind from the perusal of Prof W. Thomson's papers "On a Mechanical 
Representation of Electric, Magnetic and Galvanic Forces," Cambridge and 
Dublin Mathematical Journal, January, 1847, and his "Mathematical Theory of 
Magnetism," Philosophical Transactions, Part I. 1851, Art. 78, &c. As an 
instance of the help which may be derived from other physical investigations, 
I may state that after I had investigated the Theorems of this paper 
Professor Stokes pointed out to me the use which he had made of similar 
expressions in his "Dynamical Theory of Diffraction," Section 1, Camhndge 
Transactions, Vol. ix. Part 1. Whether the theory of these functions, consi- 
dered with reference to electricity, may lead to new mathematical ideas to be 
employed in physical research, remains to be seen. I propose in the rest of 
this paper to discuss a few electrical and magnetic problems with reference to 
spheres. These are intended merely as concrete examples of the methods of 
which the theory has been given ; I reserve the detailed investigation of cases 
chosen with special reference to experiment till I have the means of testing 
their results. 



Examples. 
I. Theory of Electrical Images. 

The method of Electrical Images, due to Prof W. Thomson"", by whicli 
the theory of spherical conductors has been reduced to great geometrical sim- 
plicity, becomes even more simple when we see its connexion with the methods 
of this paper. We have seen that the pressure at any point in a uniform 
medium, due to a spherical shell (radius = a) giving out fluid at the rate of 

a" 
AnPa^ units in unit of time, is ^P— outside the shell, and kPa inside it, 

r 

where r is the distance of the point from the centre of the shell. 

* See a series of papers "On the Mathematical Theory of Electricity," in the Cambridge and 
Dublin Math. Jour., beginning March, 1848. 

VOL L 27 



210 ON Faraday's lines of force. 

If there be two shells, one giving out fluid at a rate inPa\ and the 
other absorbing at the rate of iirFa\ then the expression for the pressure will 
be, outside the shells, 

J^ r r 

where r and / are the distances from the centres of the two shells. Equating 
this expression to zero we have, as the surface of no pressure, that for which 



/ _ Fa'' 
r ~ Pa' 



Now the surface, for which the distances to two fixed points hav^e a given 

ratio, is a sphere of which the centre is in the line joining the centres of 
the shells CC produced, so that 

and its radius ^ ^ 



Pa'lt-F^' 

If at the centre of this sphere we place another source of the fluid, then 
the pressure due to this source must be added to that due to the other two; 
and since this additional pressure depends only on the distance from the centre, 
it will be constant at the surface of the sphere, where the pressure due to 
the two other sources is zero. 

We have now the means of arranging a system of sources within a given 
sphere, so that when combined with a given system of sources outside the 
sphere, they shall produce a given constant pressure at the surface of the sphere. 

Let a be the radius of the sphere, and p the given pressure, and let the 
given sources be at distances 6„ h„ &c. from the centre, and let their rates of 
production be 4.TrP„. 47rP„ &c. 

Then if at distances ^ , ?- , &c. (measured in the same direction as h„ \, &c. 
from the centre) we place negative sources whose rates are 

-47rP,?, -477P,^, &c., 

0, Oj 



ON Faraday's lines of force. 211 

the pressure at the surface r = a will be reduced to zero. Now placing a source 
477-^ at the centre, the pressure at the surface will be uniform and equal to />. 

The whole amount of fluid emitted by the surface r = a may be found by 
adding the rates of production of the sources within it. The result is 

To apply this result to the case of a conducting sphere, let us suppose 
the external sources inP^, AnP^ to be small electrified bodies, containing e„ e, 
of positive electricity. Let us also suppose that the whole charge of the con- 
ducting sphere is =E previous to the action of the external points. Then all 
that is required for the complete solution of the problem is, that the surface 
of the sphere shall be a surface of equal potential, and that the total charge 
of the surface shall be E. 

If by any distribution of imaginary sources within the spherical surface we 
can effect this, the value of the corresponding potential outside the sphere is 
the true and only one. The potential inside the sphere must really be constant 
and equal to that at the surface. 

We must therefore find the images of the external electrified points, that 
is, for every point at distance b from the centre we must find a point on the 

same radius at a distance j- , and at that point we must place a quantity 
= — e , of imaginary electricity. 

At the centre we must put a quantity E' such that 
K = E + e,^ + e,^- + kc.; 

then if i^ be the distance from the centre, r„ r^, &c. the distances from the 
electrified points, and r\, r\, &c. the distances from their images at any point 
outside the sphere, the potential at that point will be 

E e, ( a \ ci\ e, /a b, a\ , . 



212 ON Faraday's lines of force. 

This is the value of the potential outside the sphere. At the surface we 



have 



K = a and — = -7- , — = -7- , ac. 



so that at the surface 






and this must also be the value oi p for any point within the sphere. 

For the application of the principle of electrical images the reader is referred 
to Prof Thomson's papers in the Cambridge and Dublin Mathematical Journal. 

The only case which we shall consider is that in which A = /, and b^ is infi- 
nitely distant along the axis of x, and j&=0. 

The value p outside the sphere becomes then 

and inside ^ = 0. 



II. On the effect of a paramagnetic or diam/xgnetic sphere in a uniform field oj 

magnetic force'^. 

The expression for the potential of a small magnet placed at the origin of 
co-ordinates in the direction of the axis of x is 



dx \rj~ 



i:^i'-]=-lm^ 



The eflPect of the sphere in disturbing the lines of force may be supposed 
as a first hypothesis to be similar to that of a small magnet at the origin, 
whose strength is to be determined. (We shall find this to be accurately true.) 

* See Prof. Thomson, on the Theory of Magnetic Induction, PhiL Mag. March, 1851. The induc- 
tive capadiy of the sphere, according to that paper, is the ratio of the qv/iTiiUy of magnetic induction 

(not the intensity) within the sphere to that without It is therefore equal to j^T = 2k k ' ^^^^^' 
ing to our notation. 



ON Faraday's lines of force. 213 

Let the value of the potential undisturbed by the presence of the sphere be 

'p = Ix. 
Let the sphere produce an additional potential, which for external points is 

, . a' 

and let the potential within the sphere be 

Pi = Bx. 

Let k' be the coefficient of resistance outside, and k inside the bphere, then 
the conditions to be fulfilled are, that the interior and exterior potentials should 
coincide at the surface, and that the induction through the surface should be the 
same whether deduced from the external or the internal potential. Putting 
a; = rcos^, we have for the external potential 

P = //r + ^^')cos^, 

and for the internal 

p^ = Brco%dy 

and these must be identical when r = a, or 

I+A = B. 

The induction through the surface in the external medium is 

and that through the interior surface is 

and .•. i(7-2^) = i£. 



These equations give 



A = ^^f^J, B= ^^ 



2k + k' ' ik + k' 



The effect outside the sphere is equal to that of a little magnet whose 
length is I and moment ml, provided 



214 



ON Faraday's lines of force. 



Suppose this uniform field to be that due to terrestrial magnetism, then, 
if k is less than k' as in paramagnetic bodies, the marked end of the equi- 
valent magnet will be turned to the north. If A; is greater than F as in 
diamagnetic bodies, the unmarked end of the equivalent magnet would be turned 
to the north. 



III. Magnetic Jield of variable Intensity. 



Now suppose the intensity in the undisturbed magnetic field to vary in 
magnitude and direction from one point to another, and that its components 
in X, y, z are represented by a, /8, y, then, if as a first approximation we re- 
gard the intensity within the sphere as sensibly equal to that at the centre, 
the change of potential outside the sphere arising from the presence of the 
sphere, disturbing the lines of force, will be the same as that due to three 
small magnets at the centre, with their axes parallel to x, y, and z, and their 
moments equal to 

k-k' 3 k-k' 5^ k-k' 

2kTk'^^' 2FFF^^' 2FfF"^- 

The actual distribution of potential within and without the sphere may be 
conceived as the result of a distribution of imaginary magnetic matter on the 
surface of the sphere ; but since the external effect of this superficial magnetism 
is exactly the same as that of the three small magnets at the centre, the 
mechanical effect of external attractions will be the same as if the three ma^ets 
really existed. 

Now let three small magnets whose lengths are l^, k, k, and strengths 
m„ m^, m„ exist at the point x, y, z with their axes parallel to the axes of 
then resolving the forces on the three magnets in the direction of X, we 



X, y, z 
have 



X = 'm^ 



da Zi 



•a + 



da l^ 
dx 2 



Y +'in.-{ 



a + 



a + 



da I, 
dy 2 

da It 

dy2\ 



■+«i. 



da /g" 



a + 



da Zj 
dz 2. 



J da T da , da 



ON Faraday's lines of force. 215 



Substituting the values of the moments of the imaginary magnets 
J da ^(IB dy\ k-k' a' d , , r>^ , 2\ 



2k + k' 

The force impelling the sphere in the direction of x is therefore dependent 
on the variation of the square of the intensity or (a' + ^ + y), as we move along 
the direction of x, and the same is true for y and z, so that the law is, that 
the force acting on diamagnetic spheres is from places of greater to places of 
less intensity of magnetic force, and that in similar distributions of magnetic 
force it varies as the mass of the sphere and the square of the intensity. 

It is easy by means of Laplace's CoeflBcients to extend the approximation 
to the value of the potential as far as we please, and to calculate the attrac- 
tion. For instance, if a north or south magnetic pole whose strength is M, be 
placed at a distance b from a diamagnetic sphere, radius a, the repulsion will be 

When r is small, the first term gives a sufficient approximation. The repul- 



sion is then as the square of the strength of the pole, and the mass of the 
sphere directly and the fifth power of the distance inversely, considering the 
pole as a point. 



IV. Tivo Spheres in uniform jield. 

Let two spheres of radius a be connected together so that their centres are 
kept at a distance h, and let them be suspended in a uniform magnetic field, 
then, although each sphere by itself would have been in equilibrium at any part 
of the field, the disturbance of the field will produce forces tending to make the 
balls set in a particular direction. 

Let the centre of one of the spheres be taken as origin, then the undis- 
turbed potential is 

p = Ir cos dy ■ 

and the potential due to the sphere is 

^ k — k' a? a 



216 



ON Faraday's lines of force. 



The whole potential is therefore equal to 

l(r + 



'2jc + k' 



dp 
dr 



,^003 0= p.. 



dp 
dr 



\ldp 



Idp 
rdS 

1 



dp\ 






|=«- 



T^^m'Bdi 



^'{i+^'^*(i-3-«''')+i5r^'(i+3-''')} 



This is the value of the square of the intensity at any point. The moment 
of the couple tending to turn the combination of balls in the direction of the 
original force 

L = l^a^i7;fn?<n when r = h, 



dd \2k + k' 



L^^P 



k-k' 



2k-\-k' 



k — k' a\ . ^^ 



This expression, which must be positive, since h is greater than a, gives the 
moment of a force tending to turn the line joining the centres of the spheres 
towards the original lines of force. 

Whether the spheres are magnetic or diamagnetic they tend to set in the 
axial direction, and that without distinction of north and south. If, however, 
one sphere be magnetic and the other diamagnetic, the line of centres will set 
equatoreally. The magnitude of the force depends on the square of (k — k'), and 
is therefore quite insensible except in iron*. 



V. Two Spheres between the poles of a Magnet. 

Let us next take the case of the same balls placed not in a uniform field 
but between a north and a south pole, ±M, distant 2c from each other in the 
direction of x. 



* See Prof. Thomson in Phil. Mag. March, 1851. 



ON Faraday's lines of force. 217 

The expression for the potential, the middle of the line joining the poles 
being the origin, is 

p=m(, ' —, ' )■ 

Wc* + i^-2crcos0 Vc' + ?-' + 2crcos^/ 
From this we find as the value of P, 

P = i^7l_3!:+9^,cos-<^): 

c* \ C^ & ] 

.'. I~= - 18 ^^V sin 2^. 

and the moment to turn a pair of spheres (radius a, distance 2h) in the 
direction in which is increased is 

-^'wvk'-^''''^^' 

This force, which tends to turn the line of centres equatoreally for diamagnetic 
and axially for magnetic spheres, varies directly as the square of the strength of 
the magnet, the cube of the radius of the spheres and the square of the dis- 
tance of their centres, and inversely as the sixth power of the distance of the 
poles of the magnet, considered as points. As long as these poles are near each 
other this action of the poles will be much stronger than the mutual action of 
the spheres, so that as a general rule we may say that elongated bodies set 
axially or equatoreally between the poles of a magnet according as they are mag- 
netic or diamagnetic. If, instead of being placed between two poles very near 
to each other, they had been placed in a uniform field such as that of terrestrial 
magnetism or that produced by a spherical electro-magnet (see Ex. viii.), an 
elongated body would set axially whether magnetic or diamagnetic. 

In all these cases the phenomena depend on k — k', so that the sphere con- 
ducts itself magnetically or diamagnetically according as it is more or less 
magnetic, or less or more diamagnetic than the medium in which it is placed. 



VI. On the Magnetic Phenomena of a Sphere cut from a substance whose 
coefficient of resistance is diffierent in different directions. 

Let the axes of magnetic resistance be parallel throughout the sphere, and 

let them be taken for the axes of x, y, z. Let K, k„ k„ be the coefficients of 

resistance in these three directions, and let k' be that of the external medium, 

VOL. I. 28 



218 ON FARADAY S LINES OF FORCE. 

and a the radius of the sphere. Let / be the undisturbed magnetic intensity 
of the field into which the sphere is introduced, and let its direction- cosines 
be I, m, n. 

Let us now take the case of a homogeneous sphere whose coefficient is ^, 
placed in a uniform magnetic field whose intensity is II in the direction of x. 
The resultant potential outside the sphere would be 

and for internal points 

So that in the interior of the sphere the magnetization is entirely in the direc- 
tion of X. It is therefore quite independent of the coefficients of resistance in 
the directions of x and y, which may be changed from X\ into k^ and ^3 with- 
out disturbing this distribution of magnetism. We may therefore treat the sphere 
as homogeneous for each of the three components of /, but we must use a 
different coefficient for each. We find for external points 

and for internal points 

The external effect is the same as that which would have been produced 
if the small magnet whose moments are 

te§'^"'' ^™^"'' te^'"-^"*' 

had been placed at the origin with their directions coinciding with the axes of 
Xy y, z. The effect of the original force / in turning the sphere about the axis 
of x may be found by taking the moments of the components of that force 
on these equivalent magnets. The moment of the force in the direction of y 
acting on the third magnet is 

and that of the force in z on the second magnet is 

2k^-\-k 



ON FARADAY S LINES OF FORCE. 219 

The whole couple about the axis of a; is therefore 

tending to turn the sphere round from the axis of y towards that of z. Sup- 
pose the sphere to be suspended so that the axis of x is vertical, and let / 
be horizontal, then if 6 be the angle which the axis of y makes with the 
direction of /, m = cos 6, n= — sin 0, and the expression for the moment becomes 

f TT^T^ hT}? i' \ ^'«' sin 2d, 

tending to increase 0. The axis of least resistance therefore sets axially, but 
with either end indifferently towards the north. 

Since in all bodies, except iron, the values of k are nearly the same as in 
a vacuum, the coefficient of this quantity can be but little altered by changing 
the value of k' to k, the value in space. The expression then becomes 

i^^^/Vsin2(9, 

independent of the external medium'". 



VII. Permanent magnetism in a spherical shell. 

The case of a homogeneous shell of a diamagnetic or paramagnetic substance 
presents no difficulty. The intensity within the shell is less than what it would 
have been if the shell were away, whether the substance of the shell be dia- 
magnetic or paramagnetic. When the resistance of the shell is infinite, and when 
it vanishes, the intensity within the sheU is zero. 

In the case of no resistance the entire effect of the shell on any point, 
internal or external, may be represented by supposing a superficial stratum of 

♦ Taking the more general case of magnetic induction referred to in Art. (28), we find, in the 
expression for the moment of the magnetic forces, a constant term depending on T, besides those 
terms which dejjend on sines and cosines of 6. The result is, that in every complete revolution in 
the negative direction round the axis of T, a certain jMJsitive amount of work is gained ; but, since 
no inexhaustible source of work can exist in nature, we must admit that T-0 in all substances, 
with resf>ect to magnetic induction. This argument does not hold in the case of electric conduction, 
or in the case of a body through which heat or electricity is passing, for such states are main- 
tained by the continual expenditure of work. See Prof Thomson, Phil. Mag. March, 1851, p. 186. 

28—2 



220 ON Faraday's lines of force. 

magnetic matter spread over the outer surface, the density being given by the 
equation 

p = 3/ cos d. 
Suppose the shell now to be converted into a permanent magnet, so that the 
distribution of imaginary magnetic matter is invariable, then the external poten- 
tial due to the shell will be 

p = —I—CO3 0, 

and the internal potential Pi— ~ ^*' ^^^ 0. 

Now let us investigate the eflfect of filling up the shell with some substance 
of which the resistance is k, the resistance in the external medium being k". 
The thickness of the magnetized shell may be neglected. Let the magnetic 
moment of the permanent magnetism be la^, and that of the imaginary super- 
ficial distribution due to the medium k = Aa\ Then the potentials are 

external p' = {I-\-A)~ cos 6, internal ^, = (/+ ^ ) r cos 0. 

The distribution of real magnetism is the same before and after the introduc- 
tion of the medium k, so that 

l/+|/=i(/+4)+|(/+^), 

The external efiect of the magnetized shell is increased or diminished according 
as A; is greater or less than k'. It is therefore increased by filling up the shell 
with diamagnetic matter, and diminished by filling it with paramagnetic matter, 
such as iron. 



VIII. Electro-magnetic spherical shell. 

Let us take as an example of the magnetic effects of electric currents, 
an electro-magnet in the form of a thin spherical sheU. Let its radius be a, 
and its thickness t, and let its external effect be that of a magnet whose 
moment is /a*. Both within and without the shell the magnetic effect may be 
represented by a potential, but within the substance of the shell, where there 



ON FARADAY S LINES OF FORCE. 221 

are electric currents, the magnetic effects cannot be represented by a potential. 
Let p', pi be the external and internal potentials, 

p' = 1 -^cosd, p^ = Ar cos 0, 

and since there is no permanent magnetism, -^ = -^- , when r = a, 

A=-2L 

If we draw any closed curve cutting the shell at the equator, and at some 
other point for which is known, then the total magnetic intensity round this 
curve will be Sla cos 0, and as this is a measure of the total electric current which 
flows through it, the quantity of the current at any point may be found by 
differentiation. The quantity which flows through the element tcW is — 3/a sin 0d0, 
so that the quantity of the current referred to unit of area of section is 

-3l^sm0. 
t 

If the shell be composed of a wire coiled round the sphere so that the number 
of coils to the inch varies as the sine of 0, then the external effect will be 
nearly the same as if the shell had been made of a uniform conducting sub- 
stance, and the currents had been distributed according to the law we have just 
given. 

If a wire conducting a current of strength /, be wound round a sphere 
of radius a so that the distance between successive coUs measured along the 

2a 
axis of cc is — , then there wiU be n coils altogether, and the value of /, for 

the resulting electro-magnet will be 

The potentials, external and internal, will be 

P=I,Q^ 003 0, p,= ■ 

The interior of the shell is therefore a uniform magnetic field. 






P =I,Q ^ cos^, p,= -21,- -cos^. 



ON FARADAY S LINES OF FORCE. 



IX. Effect of the core of the electro-magnet. 

Now let us suppose a sphere of diamagnetic or paramagnetic matter intro- 
duced into the electro-magnetic coil. The result may be obtained as in the 
last case, and the potentials become 

., J n Zk' a? ^ .J. n Sk r 

The external effect is greater or less than before, according as yfc' is greater 
or less than k, that is, according as the interior of the sphere is magnetic or 
diamagnetic with respect to the external medium, and the internal effect is 
altered in the opposite direction, being greatest for a diamagnetic medium. 

This investigation explains the effect of introducing an iron core into an 
electro-magnet. If the value of k for the core were to vanish altogether, the 
effect of the electro-magnet would be three times that which it has without 
the core. As k has always a finite value, the effect of the core is less than this. 

In the interior of the electro-magnet we have a uniform field of magnetic 
force, the intensity of which may be increased by surrounding the coil with a 
shell of iron. If k' = 0, and the shell infinitely thick, the effect on internal points 
would be tripled. 

The effect of the core is greater in the case of a cylindric magnet, and 
greatest of aU when the core is a ring of soft iron. 



X. Electro-tonic functions in spherical dectro-magnet. 

Let us now find the electro-tonic functions due to this electro-magnet. 
They will be of the form 

ao = 0, ^^ — oiZ, y^= —<»y, 

where tu is some function of r. Where there are no electric currents, we must 
have ttj, 6j, Cj each = 0, and this implies 

d /_ . doi\ ^ 

the solution of which is 



ON Faraday's lines of force. 223 

Within the shell co cannot become infinite ; therefore oi = C^ is the solution, 
and outside a must vanish at an infinite distance, so that 

is the solution outside. The magnetic quantity within the shell is found by last 
article to be 

therefore within the sphere 

Ln 1 



* 2a 3^ + ^" 

Outside the sphere we must determine w so as to coincide at the surface 
with the internal value. The external value is therefore 

= _:?> 1 a' 
^ 2a 3k + k' r' ' 

where the shell containing the currents is made up of n coils of wire, con- 
ducting a current of total quantity /j. 

Let another wire be coiled round the shell according to the same law, and 
let the total number of coils be n ; then the total electro-tonic intensity EI^ 
round the second coil is found by integrating 

EI^ = I (oa sin 6ds, 



-i: 



along the whole length of the wire. The equation of the wire is 

/, <^ 

cos = -y- . 

nv 

where n' is a large number; and therefore 

ds = a sin 6d<^, 

= — ariTT sin- Odd, 

T?T ^'"' 2 / 27r ,j 1 

.*. EI^= -— (oan = — — ann 1 



3 """ "" 3 '"""^ 3k + k" 
E may be called the electro-tonic coeflBcient for the particular wire. 



224 ON Faraday's lines of force. 



XI. Spherical electro-magnetic CoU-Machine. 

We have now obtained the electro-tonic function which defines the action 
of the one coil on the other. The action of each coil on itself is found by- 
putting n* or n* for nn\ Let the first coil be connected with an apparatus 
producing a variable electro-motive force F. Let us find the efiects on both 
wires, supposing their total resistances to be i2 and R, and the quantity of 
the currents / and /'. 

Let N stand for -^ (sk+k") ' *^^^ *^® electro-motive force of the first 

wire on the second is 

dl 



That of the second on itself is 



Nnn , . 
at 



-^<- 



The equation of the current in the second wire is therefore 

-iyr„n'f-iyr«-f=ij'i' (i). 

The equation of the current in the first wire is 

-Nn'^^^-Nnn'§ + F=RI. (2). 

EHminating the differential coefficients, we get 

n n' ~ n* 

^^ ^[r^r] di + ^-E^^RW (^)' 

from which to find / and F. For this purpose we require to know the value 
of i^ in terms of t. 

Let us first take the case in which F is constant and / and T initially = 0. 
This is the case of an electro-magnetic coil-machine at the moment when the 
connexion is made with the galvanic trough. 



ON Faraday's lines of force. 225 

Putting ^T for ^ [ji + j^J "^^ ^^ 

The primary current increases very rapidly from to >, , and the secondary 
commences at --jy — and speedily vanishes, owing to the value of t being 
generally very small 

The whole work done by either current in heating the wire or in any other 
kind of action is found from the expression 



PRdt. 



The total quantity of current is 

^ Idt. 



f. 



For the secondary current we find 



/; 



'-"-S;. f."-m'r 



The work done and the quantity of the current are therefore the same as 
if a current of quantity F = —jrr- had passed through the wire for a time t, where 



--(^a- 



This method of considering a variable current of short duration is due to 
Weber, whose experimental methods render the determination of the equivalent 
current a matter of great precision. 

Now let the electro-motive force F suddenly cease while the current in the 
primary wire is /<, and in the secondary = 0. Then we shall have for the subse- 
quent time 

, . -^ „ /„ Rn -f 



226 ON fahaday's lines of force. 

R n 
The equivalent currents are ^I^ and ^I^ -^ — , and their duration is t. 

When the communication with the source of the current is cut off, there 
will be a change of E. This will produce a change in the value of t, so that 
if i2 be suddenly increased, the strength of the secondary current will be increased, 
and its duration diminished. This is the case in the ordiaaiy coU-machines. The 
quantity N depends on the form of the machine, and may be determined by 
experiment for a machine of any shape. 



XII. Spherical shell revolving in magnetic field. 

Let us next take the case of a revolving shell of conducting matter under 
the influence of a uniform field of magnetic force. The phenomena are explained 
by Faraday in his Experimental Researches, Series ii., and references are there 
given to previous experiments. 

Let the axis of z be the axis of revolution, and let the angular velocity 
be 6). Let the magnetism of the field be represented in quantity by /, inclined 
at an angle 6 to the direction of z, in the plane of zx. 

Let R be the radius of the spherical sheU, and T the thickness. Let the 
quantities Oj, ^o* yoj.he the electro-tonic functions at any point of space; a^, \, c„ 
«i» Aj 7i symbols of magnetic quantity and intensity; a^, h^, c„ a,, 13,, y, of 
electric quantity and intensity. Let p, be the electric tension at any point, 



^'+*a.l 



(1). 






ON Faraday's lines op roRCE. 227 

The expressions for a,, ^„ y, due to the magnetifim of the field are 

^, = 5, + 2 (2 Bin ^ - a; cos ^), 

A^, B,, Co being constants; and the velocities of the particles of the revolving 
sphere are 

dx dy dz ^ 



We have therefore for the electro-motive forces 

An dt 4iT 2 



a>=-7Z-^=-- 7^008^0)0;, 



_ 1 d^o I I n 

$,= P = — -:— 7T cos uayy, 

^* 47r dt An 2 ^' 



1 / . 



' 4n dt An 2 

Returning to equations (1), we get 

^db, dct\ dfii <^y» 



j^ (db^ _dc,\d§, _dy,^^ 
\dz dy) dz dy ' 

\dx dz I dx dz An 2 

^ /da, _ dbA ^ ^ _ ^^ ^ q 
dy dx) ' ' 



^dy dx) dy dx 

From which with equation (2) we find 

11/.. 
ttj = - 7- -7- -7 sin C/a>; 
k An A 

h, = 0, 

I 1 I . a 

C, = T T- T Sin U(OX, 

k An A 



p, = - — - loi {(x* + 2/*) cos ^ - a:s sin $]. 



228 ON Faraday's lines of force. 

These expressions would determine completely the motion of electricity in 
a revolving sphere if we neglect the action of these currents on themselves. 
They express a system of circular currents about the axis of y, the quantity 
of current at any point being proportional to the distance from that axis. 
The external magnetic effect will be that of a small magnet whose moment 

is jx—i w/sin 6, with its direction along the axis of y, so that the magnetism of 

the field would tend to turn it back to the axis of x*. 

The existence of these currents will of course alter the distribution of 
the electro-tonic functions, and so they will react on themselves. Let the 
final result of this action be a system of currents about an axis in the plane 
of xy inclined to the axis of x at an angle ^ and producing an external effect 
equal to that of a magnet whose moment is FR^. 

The magnetic inductive components within the shell are 
/i sin ^ — 2/' cos ^ in x, 
— 21' sm(f> in. y, 
/i cos 6 in 2, 

Each of these would produce its own system of currents when the sphere 
is in motion, and these would give rise to new distributions of magnetism, 
which, when the velocity is uniform, must be the same as the original distri- 
bution, 

(Ii sin 6 — 21' cos <l>) in x produces 2 t^— r ot {I^ sin 6 — 2 J' cos (f>) in y, 

T 

( — 2T sin <^) in y produces 2 , m (21' sin ^) in x ; 

IiQoad in z produces no currents. 

We must therefore have the following equations, since the state of the shell 
is the same at every instant, 

T 

Lam 6- 2r cos <f) = /, sin ^ -^ — — y (o2T sin 6 

T 

- 2/ sin <^ = -— T oj (/, sin ^- 2r cos <^), 

* The expression for p^ indicates a variable electric tension in the shell, so that cuirents might 
be collected by wires touching it at the equator and poles. 



ON FARADAY 8 LINES OF FORCE. 229 

-hence cot <^ = - j w, / = ^ , 5-,^^ /i sin 6. 



7-©" 



To understand the meanmg of these expressions let us take a particular case. 

Let the axis of the revolving shell be vertical, and let the revolution be 
from north to west. Let / be the total intensity of the terrestrial magnetism, 
and let the dip be d, then Ico3$ is the horizontal component in the direction 
of magnetic north. 

The result of the rotation is to produce currents in the shell about an 

T 

axis inclined at a small angle = tan"* ——rco to the south of magnetic west, and 

the external effect of these currents is the same as that of a magnet whose 
moment is 

i , ^"^ i?7cos d. 

The moment of the couple due to terrestrial magnetism tending to stop the 
rotation is 

2i7rk To) 



2 24tTrkY + Tq}* 



i?Pc08'^, 



and the loss of work due to this in unit of time is 
24:Trk T(o' 



2 247r^?+Pa>' 



i?P cos' d. 



This loss of work is made up by an evolution of heat in the substance of 
the shell, as is proved by a recent experiment of M. Foucault (see Coniptefi 
Rendus, XLi. p. 450). 



[From the Transacti&M of the Royal Scottish Society of Arts, VoL iv. Part rv.] 



IX. Description of a New Form of the Platometer, an Instrument for 
measuring the Areas of Plane Figures drawn on Paper*. 



1. The measurement of the area of a plane figure on a map or plan is an 
operation so frequently occurring in practice, that any method by which it may 
be easily and quickly performed is deserving of attention. A very able expo- 
sition of the principle of such instruments will be found in the article on 
Planimeters in the Reports of the Juries of the Great Exhibition, 1851. 

2. In considering the principle of instruments of this kind, it will be most 
convenient to suppose the area of the figure measured by an imaginary straight 
line, which, by moving parallel to itself, and at the same 

time altering in length to suit the form of the area, 
accurately sweeps it out. 

Let AZ be a fixed vertical line, APQZ the boundary 
of the area, and let a variable horizontal line move 
parallel to itself firom A to Z, so as to have its extremi- 
ties, P, M, in the curve and in the fixed straight line. 
Now, suppose the horizontal line (which we shall caU the 
generating line) to move from the position PM to QNy 
MN being some small quantity, say one inch for distinct- 
ness. During this movement, the generating line will 
have swept out the narrow strip of the surface, PMNQ, 
which exceeds the portion PMNp by the smaU triangle PQp, 

But since MN, the breadth of the strip, is one inch, the strip will contain 
as many square inches as PM is inches long; so that, when the generating 




♦ Bead to the Society, 22nd Jan. 1855. 



ON A NEW FORM OF THE PLATOMETER. 



231 




line descends one inch, it sweeps out a number of square inches equal to the 
number of linear inches in its length. 

Therefore, if we have a machine with an index of any kind, which, while 
the generating line moves one inch downwards, moves forward as many degrees 
as the generating line is inches long, and if the generating line be alternately 
moved an inch and altered in length, the index will mark 
the number of square inches swept over during the whole 
operation. By the ordinary method of limits, it may be 
shown that, if these changes be made continuous instead 
of sudden, the index will still measure the area of the 
curve traced by the extremity of the generating line. 

3. When the area is bounded by a closed curve, as 
ABDC, then to determine the area we must carry the tra- 
cing point from some point A of the curve, completely round 
the circumference to A again. Then, while the tracing point 
moves from A to C, the index will go forward and mea- 
sure the number of square inches in ACRP, and, while it 
moves from C to D, the index will measure backwards the 
square inches in CRPD, so that it will now indicate the 

square inches in ACD. Similarly, during the other part of the motion from 
D to B, and from B to D, the part DBA will be measured; so that when 
the tracing point returns to D, the instrument will have measured the area 
ACDB. It is evident that the whole area will appear positive or negative 
according as the tracing point is carried round in the direction ACDB or ABDC. 

4. We have next to consider the various methods of communicating the 
required motion to the index. The first is by means of two discs, the first 
having a flat horizontal rough surface, turning on a vertical 
axis, OQ, and the second vertical, with its circumference rest- 
ing on the flat surface of the first at P, so as to be driven 
round by the motion of the first disc. The velocity of the 
second disc will depend on OP, the distance of the point of 
contact from the centre of the first disc; so that if OP be 
made always equal to the generating line, the conditions of the instrument will 
be fulfilled. 

This is accomplished by causing the index-disc to slip along the radius of 




232 ON A NEW FORM OF THE PLATOMETER. 

the horizontal disc ; so that in working the instrument, the motion of the index- 
disc is compounded of a rolling motion due to the rotation of the first disc, 
and a slipping motion due to the variation of the generating line. 

5. In the instrument presented by Mr Sang to the Society, the first disc is 
replaced by a cone, and the action of the instrument corresponds to a mathe- 
matical valuation of the area by the use of oblique co-ordinates. As he has 
himself explained it very completely, it will be enough here to say, that the 
index-wheel has still a motion of slipping as well as of rolling. 

6. Now, suppose a wheel rolling on a surface, and pressing on it with a 
weight of a pound; then suppose the coefficient of friction to be |, it will 
require a force of 2 oz. at least to produce shpping at all, so that even if the 
resistance of the axis, &c., amounted to 1 oz., the rolling would be perfect. But 
if the wheel were forcibly pulled sideways, so as to slide along in the direction 
of the axis, then, if the friction of the axis, &c., opposed no resistance to the 
turning of the wheel, the rotation would still be that due to the forward motion ; 
but if there were any resistance, however small, it would produce its effect in 
diminishing the amount of rotation. 

The case is that of a mass resting on a rough surface, which requires a 
great force to produce the shghtest motion; but when some other force acts 
on it and keeps it in motion, the very smallest force is sufficient to alter that 
motion in direction. 

7. This effect of the combination of slipping and rolling has not escaped 
the observation of Mr Sang, who has both measured its amount, and shown how 
to eliminate its effect. In the improved instrument as constructed by him, I 
believe that the greatest error introduced in this way does not equal the ordi- 
nary errors of measurement by the old process of triangulation. This accuracy, 
however, is a proof of the excellence of the workmanship, and the smoothness 
of the action of the instrument; for if any considerable resistance had to be 
overcome, it would display itself in the results. 

8. Having seen and admired these instruments at the Great Exhibition in 
1851, and being convinced that the combination of shpping and roUing was a 
drawback on the perfection of the instrument, I began to search for some 
arrangement by which the motion should be that of perfect rolling in every 



ON A NEW FORM OF TUE PLATOMETER. 233 

motion of which the instrument is capable. The forms of the rolUng parts which 
I considered were — 

1. Two equal spheres. 

2. Two spheres, the diameters being as 1 to 2. 

3. A cone and cylinder, axes at right angles. 

Of these, the first combination only suited my purpose. I devised several modes 
of mounting the spheres so as to make the principle available. That which I 
adopted is borrowed, as to many details, from the instruments already con- 
structed, so that the originality of the device may be reduced to this principle — 
The abolition of sUpping by the use of two equal spheres. 

9. The instrument (Fig. 1) is mounted on a frame, which rolls on the two 
connected wheels, MM, and is thus constrained to travel up and down the 
paper, moving parallel to itself 

CH is a horizontal axis, passing through two supports attached to the 
frame, and carrying the wheel K and the hemisphere LAP. The wheel K rolls 
on the plane on which the instrument travels, and communicates its motion to 
the hemisphere, which therefore revolves about the axis AH with a velocity 
proportional to that with which the instrument moves backwards or forwards. 

FCO is a framework (better seen in the other figures) capable of revolving 
about a vertical axis, Cc, being joined at C and c to the frame of the instru- 
ment. The parts CF and CO are at right angles to each other and horizontal. 
The part CO carries with it a ring, SOS, which turns about a vertical axis Oo. 
This ring supports the index-.sphere Bh by the extremities of its axis Ss, just 
as the meridian circle carries a terrestrial globe. By this arrangement, it will 
be seen that the axis of the sphere is kept always horizontal, while its centre 
moves so as to be always at a constant distance from that of the hemisphere. 
This distance must be adjusted so that the spheres may always remain in con- 
tact, and the pressure at the point of contact may be regulated by means of 
springs or compresses at and o acting in the direction OC, oc. In this way 
the rotation of the hemisphere is made to drive the index-sphere. 

10. Now, let us consider the working of the instrument. Suppose the arm 
CE placed so as to coincide with CD, then 0, the centre of the index-sphere 
will be in the prolongation of the axis HA. Suppose also that, when in this 
position, the equator hB of the index-sphere is in contact with the pole A of 
the hemisphere. Now, let the arch be turned into the position CE as in the 



234 ON A NEW FORM OF THE PLATOMETER. 

figure, then the rest of the framework will be turned through an equal angle, 
and the index-sphere will roll on the hemisphere till it come into the position 
represented in the figure. Then, if there be no slipping, the arc AP = BP, and 
the angle ACF = BOP. 

Next, let the instrument be moved backwards or forwards, so as to turn 
the wheel Kk and the hemisphere LI, then the index-sphere will be turned 
about its axis Ss by the action of the hemisphere, but the ratio of their veloci- 
ties will depend on their relative positions. If we draw PQ, PR, perpendiculars 
from the point of contact on the two axes, then the angular motion of the 
index-sphere will be to that of the hemisphere, as PQ is to PR; that is, as 
PQ is to QC, by the equal triangles POQ, PQC ; that is, as ED is to DC, 
by the similar triangles CQP, CDE. 

Therefore the ratio of the angular velocities is as ED to DC, but since 
DC is constant, this ratio varies as ED. We have now only to contrive some 
way of making ED act as the generating line, and the machine is complete 
(see art. 2). 

11. The arm CF is moved in the following manner: — Tt is a rectangular 
metal beam, fixed to the frame of the instrument, and parallel to the axis AH. 
cEe is a little carriage which rolls along it, having two rollers on one side and 
one on the other, which is pressed against the beam by a spring. This carriage 
carries a vertical pin, E, turning in its socket, and having a collar above, 
through which the arm CF works smoothly. The tracing point G is attached 
to the carriage by a jointed frame eGe, which is so arranged that the point 
may not bear too heavily on the paper. 

12. When the machine is in action, the tracing point is placed on a point 
in the boundary of the figure, and made to move round it always in one 
direction till it arrives at the same point again. The up-and-down motion of 
the tracing point moves the whole instrument over the paper, turns the wheel 
K, the hemisphere LI, and the index-sphere Bh ; while the lateral motion of 
the tracing point moves the carriage E on the beam Tt, and so works the arm 
CF and the framework CO; and so changes the relative velocities of the two 
spheres, as has been explained, 

13. In this way the instrument works by a perfect rolling motion, in what- 
ever direction the tracing point is moved; but since the accuracy of the result 
depends on the equality of the arcs AP and BP, and since the smallest error 



ON A NEW FORM OF THE PLATOMETER. 235 

of adjustment would, in the course of time, produce a considerable deviation 
from this equality, some contrivance is necessary to secure it. For this purpose 
a wheel is fixed on the same axis with the ring SOs, and another of the same 
size is fixed to the frame of the instrument, with its centre coinciding with the 
vertical axis through C. These wheels are connected by two pieces of watch- 
spring, which are arranged so as to apply closely to the edges of the wheels. 
The first is firmly attached to the nearer side of the fixed wheel, and to the 
farther side of the moveable wheel, and the second to the farther side of the 
fixed wheel, and the nearer side of the moveable wheel, crossing beneath the 
first steel band. In this way the spheres are maintained in their proper relative 
position; but since no instrument can be perfect, the wheels, by preventing 
dei-angement, must cause some slight slipping, depending on the errors of work- 
manship. This, however, does not ruin the pretensions of the instrument, for it 
may be shown that the error introduced by slipping depends on the distance 
through which the lateral slipping takes place ; and since in this case it must 
be very small compared with its necessarily large amount in the other instru- 
ments, the error introduced by it must be diminished in the same proportion. 

14. I have shewn how the rotation of the index-sphere is proportional to 
the area of the figure traced by the tracing point. This rotation must be 
measured by means of a graduated circle attached to the sphere, and read oti" 
by means of a vernier. The result, as measured in degrees, may be interpreted 
in the following manner : — 

Suppose the instrument to be placed with the arm CF coinciding with CD, 
the equator Bh of the index-sphere touching the pole A of the hemisphere, and 
the index of the vernier at zero : then let these four operations be performed : — 

(1) Let the tracing point be moved to the right till DE = DC, and there- 
fore DCE, ACP, and F0B = A5\ 

(2) Let the instrument be rolled upwards till the wheel K has made a 
complete revolution, carrying the hemisphere with it ; then, on account of the 
equality of the angles SOP, PC A, the index-sphere will also make a complete 
revolution. 

(3) Let the arm CF be brought back again till F coincides with D. 

(4) Let the instrument be rolled back again through a complete revolution 
of the wheel K. The index-sphere will not rotate, because the point of contact 
is at the pole of the hemisphere. 



236 ON A NEW FORM OF THE PLATOMETER. 

The tracing point has now traversed the boundary of a rectangle, whose 
length is the circumference of the wheel A", and its breadth is equal to CD; 
and during this operation, the index-sphere has made a complete revolution, 
360" on the sphere, therefore, correspond to an area equal to the rectangle con- 
tained by the circumference of the wheel and the distance CD. The size of 
the wheel K being known, different values may be given to CD, so as to make 
the instrument measure according to any required scale. This may be done, 
either by shifting the position of the beam Tt, or by having several sockets 
in the carriage E for the pin which directs the arm to work in. 

15. If I have been too prolix in describing the action of an instrument 
which has never been constructed, it is because I have myself derived great 
satisfaction from following out the mechanical consequences of the mathematical 
theorem on which the truth of this method depends. Among the other forms 
of apparatus by which the action of the two spheres may be rendered available, 
is one which might be found practicable in cases to which that here given 
would not apply. In this instrument (Fig. 4) the areas are swept out by a 
radius- vector of variable length, turning round a fixed point in the plane. The 
area is thus swept out with a velocity varying as the angular velocity of the 
radius-vector and the square of its length conjointly, and the construction of the 
machine is adapted to the case as follows : — 

The hemisphere is fixed on the top of a vertical pillar, about which the rest 
of the instrument turns. The index-sphere is supported as before by a ring and 
framework. This framework turns about the vertical pillar along with the tra- 
cing point, but has also a motion in a vertical plane, which is communicated to 
it by a curved slide connected with the tracing point, and which, by means of a 
prolonged arm, moves the framework as the tracing point is moved to and from 
the pillar. 

The form of the curved slide is such, that the tangent of the angle of 
inclination of the line joining the centres of the spheres with the vertical is 
proportional to the square of the distance of the tracing point from the vertical 
axis of the instrument. The curve which fulfils this condition is an hyperbola, 
one of whose asymptotes is vertical, and passes through the tracing point, and 
the other horizontal through the centre of the hemisphere. 

The other parts of this instrument are identical with those belonging to 
that alreadv described. 



VOL. /. PLATE n. 




Fig.iFlan 



C %K 




FigJF'runr EleuaCion 



VOL. I. PLATE U. 




Ti^.4. 



ON A NEW FORM OF THE PLATOMETER. 237 

When the tracing point is made to traverse the boundary of a plane figure, 
there is a continued rotation of the radius-vector combined with a change of 
length. The rotation causes the index-sphere to roll on the fixed hemisphere, 
while the length of the radius-vector determines the rate of its motion about its 
axis, so that its whole motion measures the area swept out by the radius-vector 
during the motion of the tracing point. 

The areas measured by this instrument may either lie on one side of the 
pillar, or they may extend all round it. In either case the action of the 
instrument is the same as in the ordinary case. In this form of the instrument 
we have the advantages of a fixed stand, and a simple motion of the tracing 
point; but there seem to be difficulties in the way of supporting the spheres 
and arranging the shde ; and even then the instrument would require a tall 
pillar, in order to take in a large area. 

16. It will be observed that I have said little or nothing about the prac- 
tical details of these instruments. Many useful hints will be found in the large 
work on Platometers, by Professor T. Gonnellu, who has given us an account 
of the difficulties, as well as the results, of the construction of his most 
elaborate instrument. He has also given some very interesting investio-ations 
into the errors produced by various irregularities of construction, although, as 
far as I am aware, he has not even suspected the error which the sliding of 
the index-wheel over the disc must necessarily introduce. With respect to this, 
and other points relating to the working of the instrument, the memoir of 
Mr Sang, in the Transactions of this Society, is the most complete that I 
have met with. It may, however, be as well to state, that at the time when 
I devised the improvements here suggested, I had not seen that paper, though 
I had seen the instrument standing at rest in the Crystal Palace. 

Edinburgh, 30th January, 1855, 
Note. — Since the design of the above instrument was submitted to the Society of Arts, 
I have met with a description of an instrument combining simplicity of construction with 
the power of adaptation to designs of any size, and at the same time more portable than 
any other instrument of the kind. Althougli it does not act by perfect rolling, and there- 
fore belongs to a different class of instruments from that described in this paper, I think 
that its simplicity, and the beauty of the principle on which it acts, render it worth the 
attention of engineers and mechanists, whether practical or theoretical. A full account of 
this instrument is to be found in Moigno's " Cosmos," 5th year, Vol. viii., Part viii., p. 213, 
published 20th February 1856. Description et Theorie du planiniHre polaire, invents par 
J. Amsler, de Schaffuuse en Suisse. 
Cambridge, 30th April, 1856. 



[From the Cambridge Philosophical Society Proceedings, Vol. i. pp. 173 — 175.] 



X. 0?i the Elementary TJieory of Optical Instruments. 

The object of this communication was to shew how the magnitude and 
position of the image of any object seen through an optical instrument could 
be ascertained without knowing the construction of the instrument, by means 
of data derived from two experiments on the instrument. Optical questions 
are generally treated of with respect to the pencils of rays which pass through 
the instrument. A pencil is a collection of rays which have passed through one 
point, and may again do so, by some optical contrivance. Now if we suppose 
all the points of a plane luminous, each will give out a pencil of rays, and 
that collection of pencils which passes through the instrument may be treated 
as a beam of hght. In a pencil only one ray passes through any point of 
space, unless that point be the focus. In a beam an infinite number of rays, 
corresponding each to some point in the luminous plane, passes through any 
point; and we may, if we choose, treat this collection of rays as a pencil 
proceeding from that point. Hence the same beam of light may be decomposed 
into pencils in an infinite variety of ways; and yet, since we regard it as the 
same collection of rays, we may study its properties as a beam independently 
of the particular way in which we conceive it analysed into pencils. 

Now in any instrument the incident and emergent beams are composed 
of the same light, and therefore every ray in the incident beam has a 
corresponding ray in the emergent beam. We do not know their path within 
the instrument, but before incidence and after emergence they are straight 
lines, and therefore any two points serve to determine the direction of each. 

Let us suppose the instrument such that it forms an accurate image of a 
plane object in a given position. Then every ray which passes through a given 



ON THE ELEMENTARY THEORY OF OPTICAL INSTRUMENTS. 239 

point of the object before incidence passes through the corresponding point of 
the image after emergence, and this determines one point of the emergent ray. 
If at any other distance from the instrument a plane object has an accurate 
image, then there will be two other corresponding points given in the incident 
and emergent rays. Hence if we know the points in which an incident ray 
meets the planes of the two objects, we may find the incident ray by joining 
the points of the two images corresponding to them. 

It was then shewn, that if the image of a plane object be distinct, flat, and 
similar to the object for two different distances of the object, the image of any 
other plane object perpendicular to the axis will be distinct, flat and similar 
to the object. 

When the object is at an infinite distance, the plane of its image is the 
principal focal plane, and the point where it cuts the axis is the piincipal 
focus. The line joining any point in the object to the corresponding point of 
the image cuts the axis at a fixed point called the focal centre. The distance 
of the principal focus from the focal centre is called the principal focal length, 
or simply the focal length. 

There are two principal foci, etc., formed by incident parallel rays passing 
in opposite directions through the instrument. If we suppose light always to 
pass in the same direction through the instrument, then the focus of incident 
rays when the emergent rays are parallel is the Jirst principal focus, and the 
focus of emergent rays when the incident rays are parallel is the second 
principal focus. 

Corresponding to these we have first and second focal centres and focal 
lengths. 

Now let Q, be the focus of incident rays, P^ the foot of the perpendicular 
from ^1 on the axis, Q, the focus of emergent rays, P, the foot of the corre- 
sponding perpendicular, F^F^ the first and second principal foci, A^A^ the first and 
second focal centres, then 

F\F\ _PjQr_FJP, 

A^Frp.QrFA.' 

lines being positive when measured in the direction of the light. Therefore 
the position and magnitude of the image of any object is found by a simple 
proportion. 



240 ON THE ELEMENTARY THEORY OF OPTICAL INSTRUMENTS. 

In one important class of instruments there are no principal foci or focal 
centres. A telescope in which parallel rays emerge parallel is an instance. In 
such instruments, if m be the angular magnifying power, the linear dimensions 

of the image are — of the object, and the distance of the image of the object 

from the image of the object-glass is —^ of the distance of the object from 

the object-glass. Rules were then laid down for the composition of instruments, 
and suggestions for the adaptation of this method to second approximations, and 
the method itself was considered with reference to the labours of Cotes, Smith, 
Euler, Lagrange, and Gauss on the same subject. 



[From the Report of the British Association, 1856.] 



XI. On a Method of Drawing the Theoi-etical Forms of Faraday s Lines of 
Force without Calculation. 



The method applies more particularly to those cases in which the lines 
are entirely parallel to one plane, such as the lines of electric currents in a 
thin plate, or those round a system of parallel electric currents. In such cases, 
if we know the forms of the lines of force in any two cases, we may combine 
them by simple addition of the functions on which the equations of the lines 
depend. Thus the system of lines in a uniform magnetic field is a series of 
parallel straight lines at equal intervals, and that for an infinite straight electric 
current perpendicular to the paper is a series of concentric circles whose radii 
are in geometric progression. Having drawn these two sets of lines on two 
separate sheets of paper, and laid a third piece above, draw a third set of lines 
through the intersections of the first and second sets. This will be the system 
of lines in a uniform field disturbed by an electric current. The most interesting 
cases are those of uniform fields disturbed by a small magnet. If %ve draw a 
circle of any diameter with the magnet for centre, and join those points in which 
the circle cuts the lines of force, the straight lines so drawn will be parallel and 
equidistant; and it is easily shown that they represent the actual lines of 
force in a paramagnetic, diamagnetic, or crystallized body, according to the 
nature of the original lines, the size of the circle, &c. No one can study 
Faraday's researches without wishing to see the forms of the Hnes of force. 
This method, therefore, by which they may be easily drawn, is recommended 
to the notice of electrical students. 



[From the Report of the British Association, 1856.] 



XII. On the Unequal Sensibility of the Foramen Centrale to Light of 
different Colours. 

When observing tlie spectrum formed by looking at a long vertical slit 
through a simple prism, I noticed an elongated dark spot running up and down 
in the blue, and following the motion of the eye as it moved up and down 
the spectrum, but refusing to pass out of the blue into the other colours. It 
was plain that the spot belonged both to the eye and to the blue part of the 
spectrum. The result to which I have come is, that the appearance is due to 
the yellow spot on the retina, commonly called the Foramen Centrale of Soem- 
mering. The most convenient method of observing the spot is by presenting 
to the eye in not too rapid succession, blue and yellow glasses, or, still better, 
allowing blue and yellow papers to revolve slowly before the eye. In this way 
the spot is seen in the blue. It fades rapidly, but is renewed every time the 
yellow comes in to relieve the effect of the blue. By using a Nicol's prism 
along with this apparatus, the brushes of Haidinger are well seen in connexion 
with the spot, and the fact of the brushes being the spot analysed by polarized 
light becomes evident. If we look steadily at an object behind a series of bright 
bars which move in front of it, we shall see a curious bending of the bars as 
they come up to the place of the yellow spot. The part which comes over the 
spot seems to start in advance of the rest of the bar, and this would seem to 
indicate a greater rapidity of sensation at the yellow spot than in the surround- 
ing retina. But I find the experiment diflScult, and I hope for better results 
from more accurate observers. 



[From the Report of the British Association, 1856.] 



XIII. On the TJieory of Compound Colours with reference to Mixtures of Blue 

and Yellow Light. 

When we mix together blue and yellow paint, we obtain green paint. This 
fact is well known to all who have handled colours ; and it is universally 
admitted that blue and yellow make green. Red, yellow, and blue, being the 
primary colours among painters, green is regarded as a secondary colour, arising 
from the mixture of blue and yellow. Newton, however, found that the green 
of the spectrum was not the same thing as the mixture of two colours of the 
spectrum, for such a mixture could be separated by the prism, while the green 
of the specti-um resisted further decomposition. But still it was believed that 
yellow and blue would make a green, though not that of the spectrum. As 
far as I am aware, the first experiment on the subject is that of M. Plateau, 
who, before 1819, made a disc with alternate sectors of prussian blue and gam- 
boge, and observed that, when spinning, the resultant tint was not green, but 
a neutral gray, inclining sometimes to yellow or blue, but never to green. Prof 
J. D, Forbes of Edinburgh made similar experiments in 1849, with the same 
result. Prof Helmholtz of Konigsberg, to whom we owe the most complete 
investigation on visible colour, has given the true explanation of this phaenomenon. 
The result of mixing two coloured powders is not by any means the same as 
mixing the beams of light which flow from each separately. In the latter case 
we receive all the light which comes either from the one powder or the other. 
In the former, much of the light coming from one powder falls on particles of 
the other, and we receive only that portion which has escaped absorption by one 
or other. Thus the light coming from a mixture of blue and yellow powder, 
consists partly of light coming directly from blue particles or yellow particles, 
and partly of light acted on by both blue and yellow particles. This latter light 
is green, since the blue stops the red, yellow, and orange, and the yellow stops 



244 ON THE THEORY OF COMPOUND COLOURS. 

the blue and violet I have made experiments on the mixture of blue and 
vellow light — by rapid rotation, by con\bined reflexion and transmission, by view- 
ing them out of focus, in stripes, at a gre;it distiince, by throwing the colours 
of the spectrum on a screen, and by receiving them into the eye directly ; and 
I have arranged a portable apparatus by which any one may see the result of 
this or any other mLxture of the colours of the spectrum. In all these cases 
blue and yellow do not make green. I have also made experiments on the 
mixture of coloured powders. Those which I used principally were "mineral 
blue" (from copper) and "chrome-yellow." Other blue and yellow pigments gave 
curious results, but it was more difficult to make the mixtures, and the greens 
were less uniform in tint. The mixtures of these colours were made by weight, 
and were painted on discs of paper, which were afterwards treated in the manner 
described in my paper " On Colour as perceived by the Eye," in the Transactions 
of the Boyal Soi.'icti/ of Edinburgh, Vol. xxi. Part 2. The \'isible effect of the 
colour is estimated in terms of the standard-coloured papers : — vermilion (V), 
ultramarine (U), and emerald-green (E). The accmucy of the results, and their 
sijjnificance, can be best understood by referring to the paper before mentioned. 
I shall denote mineral blue by B, and chrome-yellow by Y ; and B, Y, means 
a mixture of three parts blue and five parts yellow. 

Given Colour. Standard Colours. Coefficient 

V. U. E. of brightness. 

B, , 100 = 2 36 7 45 

B- Y, , 100 = 1 18 17 37 

B. Y, , 100 = 4 11 34 49 

B, Y, , 100 =9 5 40 54 

B, Y. , 100 = 15 1 40 56 

B, Y, , 100 = 22 - 2 44 64 

B, Y. , 100 = 35-10 51 76 

B, Y, , 100 = 64-19 64 109 

Y, , 100 = 180 -27 124 277 

The columns Y, U, E give the proportions of the standard colours which 
are equivalent, to 100 of the given colour; and the sum of V, U, E gives a co- 
efficient, which gives a general idea of the brightness. It will be seen that the 
tirst admixture of yellow diminishes the brightness of the blue. The negative 
vidues of U indicate that a mixture of Y, U, and E cannot be made equivalent 
to the given colour. The experiments from which these results were taken had 



ON THE THEORY OF COMPOUND COLOURS. 245 

the negative values tran-sferred to the other side of the equation. They were 
all made by means of the colour-top, and were verified by repetition at different 
times. It may be necessary to remark, in conclusion, with reference to the mode 
of registering visible colours in terms of three arbitrary standard colours, that it 
proceeds upon that theory of three primary elements in the sensation of colour, 
which treats the investigation of the laws of visible colour as a bmnch of human 
physiology, incapable of being deduced from the laws of light itself, as set forth 
in physical optics. It takes advantage of the methods of optics to study vision 
itself; and its appeal is not to physical principles, but to our consciousness of 
our own sensations. 



[From the Report of ike British Association, 1856.] 



XIV. On an Instrument to illxLstrate Poinsdt's Theory of Rotation. 

In studying the rotation of a solid body according to Poinsdt's method, we 
have to consider the successive positions of the instantaneous axis of rotation 
with reference both to directions fixed in space and axes assumed in the moving 
body. The paths traced out by the pole of this axis on the invariable plane and 
on the central ellipsoid form interesting subjects of mathematical investigation. 
But when we attempt to follow with our eye the motion of a rotating body, 
we find it difficult to determine through what point of the body the instantaneous 
axis passes at any time, — and to determine its path must be still more difficult. 
I have endeavoured to render visible the path of the instantaneous axis, and to 
vary the circumstances of motion, by means of a top of the same kind as that 
used by Mr Elliot, to illustrate precession^'. The body of the instrument is a 
hoUow cone of wood, rising from a ring, 7 inches in diameter and 1 inch thick. 
An iron axis, 8 inches long, screws into the vertex of the cone. The lower 
extremity has a point of hard steel, which rests in an agate cup, and forms the 
support of the instrument. An iron nut, three ounces in weight, is made to 
screw on the axis, and to be fixed at any point; and in the wooden ring are 
screwed four bolts, of three ounces, working horizontally, and four bolts, of one 
ounce, working vertically. On the upper part of the axis is placed a disc of 
card, on which are drawn four concentric rings. Each ring is divided into four 
quadrants, which are coloured red, yellow, green, and blue. The spaces between 
the rings are white. When the top is in motion, it is easy to see in which quad- 
rant the instantaneous axis is at any moment and the distance between it and 
the axis of the instrument; and we observe, — 1st. That the instantaneous axis 
travels in a closed curve, and returns to its original position in the body. 2ndly. 

* Transactions of the Royal Scottish Society of Arts, 1855. 



ON AN INSTRUMENT TO ILLUSTRATE POINSOT S THEORY OF ROTATION. 247 

That by working the vertical bolts, we can make the axis of the instrument 
the centre of this closed curve. It will then be one of the principal axes of 
inertia. 3rdly. That, by working the nut on the axis, we can make the order 
of colours either red, yellow, green, blue, or the reverse. When the order of 
colours is in the same direction as the rotation, it indicates that the axis of the 
instrument is that of greatest moment of inertia. 4thly. That if we screw the 
two pairs of opposite horizontal bolts to different distances from the axis, the 
path of the instantaneous pole will no longer be equidistant from the axis, but 
will describe an ellipse, whose longer axis is in the direction of the mean axis 
of the instrument. 5thly. That if we now make one of the two horizontal axes 
less and the other greater than the vertical axis, the instantaneous pole will 
separate from the axis of the instrument, and the axis will incline more and more 
till the spinning can no longer go on, on account of the obliquity. It is easy 
to see that, by attending to the laws of motion, we may produce any of the 
above effects at pleasure, and illustrate many different propositions by means of 
the same instrument. 



[From the Transactions of the Royal Society of Edinburgh, Vol. xxi. Part iv.] 



XV. On a Dynamical Top, for exhibiting the phenomena of the motion of a 
system of invariable form about a fixed point, with some suggestions as to 
the Earth's mx)tion. 

(Read 20th April, 1857.) 

To those who study the progress of exact science, the common spinning-top 
is a symbol of the labours and the perplexities of men who had successfully 
threaded the mazes of the planetary motions. The mathematicians of the last 
age, searching through nature for problems worthy of their analysis, foimd in 
this toy of their youth, ample occupation for their highest mathematical powers. 

No illustration of astronomical precession can be devised more perfect than 
that presented by a properly balanced top, but yet the motion of rotation has 
intricacies far exceeding those of the theory of precession. 

Accordingly, we find Euler and D'Alembert devoting their talent and their 
patience to the estabhshment of the laws of the rotation of solid bodies. 
Lagrange has incorporated his own analysis of the problem with his general 
treatment of mechanics, and since his time M. Poins6t has brought the subject 
under the power of a more searching analysis than that of the calculus, in 
which ideas take the place of symbols, and intelligible propositions supersede 
equations. 

In the practical department of the subject, we must notice the rotatory 
machine of Bohnenberger, and the nautical top of Troughton. In the first of 
these instruments we have the model of the Gyroscope, by which Foucault has 
been able to render visible the effects of the earth's rotation. The beautiful 
experiments by which Mr J. EUiot has made the ideas of precession so familiar 
to us are performed with a top, similar in some respects to Troughton's, though 
not borrowed from his. 



ON A DYNAMICAL TOP. 249 

The top which I have the honour to spin before the Society, differs from 
that of Mr Elliot in having more adjustments, and in being designed to exhibit 
far more complicated phenomena. 

The arrangement of these adjustments, so as to produce the desired effects, 
depends on the mathematical theory of rotation. The method of exhibiting the 
motion of the axis of rotation, by means of a coloured disc, is essential to the 
success of these adjustments. This optical contrivance for rendering visible the 
nature of the rapid motion of the top, and the practical methods of applying 
the theory of rotation to such an instrument as the one before us, are the 
grounds on which I bring my instrument and experiments before the Society 
as my own. 

I propose, therefore, in the first place, to give a brief outline of such parts 
of the theory of rotation as are necessary for the explanation of the phenomena 
of the top. 

I shall then describe the instrument with its adjustments, and the effect of 
each, the mode of observing of the coloured disc when the top is in motion, and 
the use of the top in illustrating the mathematical theory, with the method of 
making the different experiments. 

Lastly, I shall attempt to explain the nature of a possible variation in the 
earth's axis due to its figure. This variation, if it exists, must cause a periodic 
inequality in the latitude of every place on the earth's surface, going through its 
period in about eleven months. The amount of variation must be very small, 
but its character gives it importance, and the necessary observations are already 
made, and only require reduction. 



On the Tlieory of Rotation. 

The theory of the rotation of a rigid system is strictly deduced from the 
elementary laws of motion, but the complexity of the motion of the particles of 
a body freely rotating renders the subject so intricate, that it has never been 
thoroughly understood by any but the most expert mathematicians. Many who 
have mastered the lunar theory have come to erroneous conclusions on this sub- 
ject ; and even Newton haa chosen to deduce the disturbance of the earth's axis 
from his theory of the motion of the nodes of a free orbit, rather than attack 
the problem of the rotation of a solid body. 



250 ON A DYNAMICAL TOP. 

The method by which M. Poinsot has rendered the theory more manageable, 
is by the liberal introduction of "appropriate ideas," chiefly of a geometrical 
character, most of which had been rendered familiar to mathematicians by the 
writings of Monge, but which then first became illustrations of this branch of 
dynamics. If any further progress is to be made in simplifying and arranging 
the theory, it must be by the method which Poins6t has repeatedly pointed out 
as the only one which can lead to a true knowledge of the subject, — that of 
proceeding from one distinct idea to another, instead of trusting to symbols and 
equations. 

An important contribution to our stock of appropriate ideas and methods has 
lately been made by Mr R. B. Hayward, in a paper, "On a Direct Method of 
estimatmg Velocities, Accelerations, and all similar quantities, with respect to axes, 
moveable in any manner in Space." {Trans. Cambridge Phil. Soc. Vol. x. Part i.) 
* In this communication I intend to confine myself to that part of the 
subject which the top is intended to illustrate, namely, the alteration of the 
position of the axis in a body rotating freely about its centre of gravity. I 
shall, therefore, deduce the theory as briefly as possible, from two considera- 
tions only, — the permanence of the original angular momentum in direction and 
magnitude, and the permanence of the original vis viva. 

•"' The mathematical difiSculties of the theory of rotation arise chiefly from 
the want of geometrical illustrations and sensible images, by which we might 
fix the results of analysis in our minds. 

It is easy to understand the motion of a body revolving about a fixed axle. 
Every point in the body describes a circle about the axis, and returns to its 
original position after each complete revolution. But if the axle itself be in 
motion, the paths of the different points of the body will no longer be circular 
or re-entrant. Even the velocity of rotation about the axis requires a careful 
definition, and the proposition that, in all motion about a fixed point, there is 
always one Hne of particles forming an instantaneous axis, is usually given in 
the form of a very repulsive mass of calculation. Most of these difficulties may 
be got rid of by devoting a little attention to the mechanics and geometry of 
the problem before entering on the discussion of the equations. 

Mr Hayward, in his paper already referred to, has made great use of the 
mechanical conception of Angular Momentum. 

* 7th May, 1857. The paragraphs marked thus have been rewritten since the paper was read. 



ON A DYNAMICAL TOP. 251 

Definition. — Jlie Angular Momentum of a particle about an axis is mea- 
sured by the product of the mass of the particle, its velocity resolved in the normal 
plane, and the perpendicular from the axis on the direction of motion. 

^' The angular momentum of any system about an axis is the algebraical 
sum of the angular momenta of its parts. 

As the rate of change of the linear momentum of a particle measures the 
moving force which acts on it, so the rate of change of angular momentum 
measures the moment of that force about an axis. 

All actions between the parts of a system, being pairs of equal and opposite 
forces, produce equal and opposite changes in the angular momentum of those 
parts. Hence the whole angular momentum of the system is not aflfected by 
these actions and re-actions. 

* When a system of invariable form revolves about an axis, the angular 
velocity of every part is the same, and the angular momentum about the axis is 
the product of the angular velocity and the moment of inertia about that axis. 

* It is only in particular cases, however, that the whole angular momentum 
can be estimated in this way. In general, the axis of angular momentum differs 
from the axis of rotation, so that there will be a residual angular momentum 
about an axis perpendicular to that of rotation, imless that axis has one of three 
positions, called the principal axes of the body. 

By referring everything to these three axes, the theory is greatly simplified. 
The moment of inertia about one of these axes is greater than that about any 
other axis through the same point, and that about one of the others is a mini- 
mum. These two are at right angles, and the third axis is perpendicular to 
their plane, and is called the mean axis. 

* Let A, B, C be the moments of inertia about the principal axes through 
the centre of gravity, taken in order of magnitude, and let Wj oj., cd^ be the 
angular velocities about them, then the angular momenta wHl be Ao)„ Bco. 
and Cwj . 

Angular momenta may be compounded like forces or velocities, by the 
law of the "parallelogram," and since these three are at right angles to each 
other, their resultant is 

JA^:^JTB%JTC^' = H (1), 

and this must be constant, both in magnitude and direction in space, since no 
external forces act on the body. 



252 ON A DYNAMICAL TOP. 

We shall call this axis of angular momentum the invariable axis. It is 

perpendicular to what has been called the invariable plane. Poins6t calls it 

the axis of the couple of impulsion. The direction-cosines of this axis in the 

body are, 

, A(o, B(o. Ca)o 

« = ^, m = -^, ^ = ^- 

Since I, m and n vary during the motion, we need some additional 
condition to determine the relation between them. We find this in the property 
of the vis viva of a system of invariable form in which there is no friction. 
The vis viva of such a system must be constant. We express this in the 
equation 

Aoj,' + B(o,'+C(o,'=V (2). 

Substituting the values of Wi, w^, Wj in terms of I, m, n, 



Let -i=a\ -T, = h\ ^=c\ 



= e' 



A ' B ' C~ ' W 
and this equation becomes 

a'Z' + 6W + cV = e» (3), 

and the equation to the cone, described by the invariable axis within the 
body, is 

(a'-e')x' + {h'-e')y'-\-{c'-e')z' = (4). 

The intersections of this cone with planes perpendicular to the principal 
axes are found by putting x, y, or z, constant in this equation. By giving 
e various values, all the different paths of the pole of the invariable axis, 
corresponding to different initial circumstances, may be traced. 

*In the figiu-es, I have supposed a' = 100, 6'= 107, and c" = 110. The 
first figure represents a section of the various cones by a plane perpendicular 
to the axis of x, which is that of greatest moment of inertia. These sections 
are ellipses having their major axis parallel to the axis of h. The value of e* 
corresponding to each of these curves is indicated by figures beside the curve. 
The ellipticity increases with the size of the ellipse, so that the section 
corresponding to 6^=107 would be two parallel straight lines (beyond the bounds 
of the figure), after which the sections would be hyperbolas. 



ON A DYNAMICAL TOP. 253 

*The second figure represents the sections made by a plane, perpendicular 
to the mean axis. They are all hyperbolas, except when 6^=107, when the 
section is two intersecting straight lines. 

The third figure shows the sections perpendicular to the axis of least 
moment of inertia. From e'=110 to ^"=107 the sections are ellipses, e*=107 
gives two parallel straight lines, and beyond these the curves are hyperbolas. 

*The fourth and fifth figures show the sections of the series of cones 

made by a cube and a sphere respectively. The use of these figures is to 

exhibit the connexion between the different curves described about the three 
principal axes by the invariable axis during the motion of the body. 

*We have next to compare the velocity of the invariable axis with respect 
to the body, with that of the body itself round one of the principal axes. 
Since the invariable axis is fixed in space, its motion relative to the body 
must be equal and opposite to that of the portion of the body through which 
it passes. Now the angular velocity of a portion of the body whose direction - 
cosines are I, m, n, about the axis of x is 

Substituting the values of w^, w^, w,, in terms of I, m, n, and taking 
account of equation (3), this expression becomes 

Changing the sign and putting 1=^tt we have the angular velocity of 
the invariable axis about that of x 

_ o>, e' — a" 

always positive about the axis of greatest moment, negative about that of least 
moment, and positive or negative about the mean axis according to the value 
of e*. The direction of the motion in every case is represented by the arrows 
in the figures. The arrows on the outside of each figure indicate the direction 
of rotation of the body, 

*If we attend to the curve described by the pole of the invariable axis 



254 ON A DYNAMICAL TOP. 

on the sphere in fig. 5, we shall see that the areas described by that point, 
if projected on the plane of yz, are swept out at the rate 

a" 
Now the semi-axes of the projection of the spherical ellipse described by 
the pole are 






Dividing the area of this ellipse by the area described during one revo- 
lution of the body, we find the number of revolutions of the body during 
the description of the ellipse — 



The projections of the spherical ellipses upon the plane of yz are all 
similar ellipses, and described in the same number of revolutions; and in each 
ellipse so projected, the area described in any time is proportional to the 
number of revolutions of the body about the axis of x, so that if we measure 
time by revolutions of the body, the motion of the projection of the pole of 
the invariable axis is identical with that of a body acted on by an attractive 
central force varying directly as the distance. In the case of the hyperbolas 
in the plane of the greatest and least axis, this force must be supposed 
repulsive. The dots in the figures 1, 2, 3, are intended to indicate roughly 
the progress made by the invariable axis during each revolution of the body 
about the axis of x, y and z respectively. It must be remembered that the 
rotation about these axes varies with their inclination to the invariable axis, 
so that the angular velocity diminishes as the inclination increases, and there- 
fore the areas in the ellipses above mentioned are not described with uniform 
velocity in absolute time, but are less rapidly swept out at the extremities of 
the major axis than at those of the minor. 

*When two of the axes have equal moments of inertia, or h — c, then 
the angular velocity (o^ is constant, and the path of the invariable axis is 
circular, the number of revolutions of the body during one circuit of the 
invariable axis, being 



ON A DYNAMICAL TOP. 255 

The motion is in the same direction as that of rotation, or in the opposite 
direction, according as the axis of x is that of greatest or of least moment 
of inertia. 

*Both in this case, and in that in which the three axes are unequal, the 
motion of the invariable axis in the body may be rendered very slow by 
dimlulshing the difference of the moments of inertia. The angular velocity of 
the axis of x about the invariable axis in space is 

to. 



a'(l-l')' 

which is greater or less than Wj, as e* is greater or less than a\ and, when 
these quantities are nearly equal, is very nearly the same as Wj itself. This 
quantity indicates the rate of revolution of the axle of the top about its 
mean position, and is very easily observed. 

*The instantaneous axis is not so easily observed. It revolves round the 
invariable axis in the same time with the axis of x, at a distance which Is very 
small in the case when a, h, c, are nearly equal. From its rapid angular motion 
in space, and Its near coincidence with the invariable axis, there Is no advantage 
in studying its motion in the top. 

*By making the moments of inertia very unequal, and in definite proportion 
to each other, and by drawing a few strong lines as diameters of the disc, the 
combination of motions will produce an appearance of epicycloids, which are the 
result of the continued intersection of the successive positions of these lines, and 
the cusps of the epicycloids lie in the curve in which the instantaneous axis 
travels. Some of the figures produced in this way are very pleasing. 

In order to illustrate the theory of rotation experimentally, we must have 
a body balanced on its centre of gravity, and capable of having Its principal 
axes and moments of inertia altered in form and position within certain limits. 
We must be able to make the axle of the instrument the greatest, least, or 
mean principal axis, or to make it not a principal axis at all, and we must be 
able to see the position of the Invariable axis of rotation at any time. There 
must be three adjustments to regulate the position of the centre of gravity, 
three for the magnitudes of the moments of inertia, and three for the directions 
of the principal axes, nine Independent adjustments, which may be distributed 
as we please among the screws of the instrument. 



256 ON A DYNAMICAL TOP. 

The form of the body of the instrument which I have found most suitable is 
that of a bell (p. 262, fig. 6). (7 is a hollow cone of brass, i2 is a heavy 
ring cast in the same piece. Six screws, with heavy heads, x, y, z, x, y', z, 
work horizontally in the ring, and three similar screws, I, m, n, work vertically 
through the ring at equal intervals. AS is the axle of the instrument, SS is 
a brass screw working in the upper part of the cone (7, and capable of being 
firmly clamped by means of the nut c. 5 is a cylindrical brass bob, which may 
be screwed up or down the axis, and fixed in its place by the nut 7). 

The lower extremity of the axle is a fine steel point, finished without emery, 
and afterwards hardened. It runs in a little agate cup set in the top of the 
pillai' P. If any emery had been embedded in the steel, the cup would soon 
be worn out. The upper end of the axle has also a steel point by which it may 
be kept steady while spinning. 

When the instrument is in use, a coloured disc is attached to the upper 
end of the axle. 

It will be seen that there are eleven adjustments, nine screws in the brass 
ring, the axle screwing in the cone, and the bob screwing on the axle. The 
advantage of the last two adjustments is, that by them large alterations can be 
made, which are not possible by means of the small screws. 

The first thing to be done with the instrument is, to make the steel point 
at the end of the axle coincide with the centre of gravity of the whole. This 
is done roughly by screwing the axle to the right place nearly, and then balancing 
the instrument on its point, and screwing the bob and the horizontal screws till 
the instrument will remain balanced in any position in which it is placed. 

When this adjustment is carefully made, the rotation of the top has no 
tendency to shake the steel point in the agate cup, however irregular the motion 
may appear to be. 

The next thing to be done, is to make one of the principal axes of the 
central ellipsoid coincide with the axle of the top. 

To effect this, we must begin by spinning the top gently about its axle, 
steadying the upper part with the finger at first. If the axle is already a 
principal axis the top will continue to revolve about its axle when the finger is 
removed. If it is not, we observe that the top begins to spin about some other 
axis, and the axle moves away from the centre of motion and then back to it 
again, and so on, alternately widening its circles and contracting them. 



ON A DYNAMICAL TOP. 257 

It is impossible to observe this motion successfully, without the aid of the 
coloured disc placed near the upper end of the axis. This disc is divided into 
sectors, and strongly coloured, so that each sector may be recognised by its colour 
when in rapid motion. If the axis about which the top is really revolving, falls 
within this disc, its position may be ascertained by the colour of the spot at the 
centre of motion. If the central spot appears red, we know that the invariable 
axis at that instant passes through the red part of the disc. 

In this way we can trace the motion of the invariable axis in the revolving 
body, and we find that the path which it describes upon the disc may be a circle, 
an ellipse, an hyperbola, or a straight line, according to the arrangement of the 
instrument. 

In the case in which the invariable axis coincides at first with the axle of 
the top, and returns to it after separating from it for a time, its true path is 
a circle or an ellipse having the axle in its circumference. The true principal 
axis is at the centre of the closed curve. It must be made to coincide with the 
axle by adjusting the vertical screws I, in, n. 

Suppose that the colour of the centre of motion, when farthest from the 
axle, indicated that the axis of rotation passed through the sector L, then the 
principal axis must also lie in that sector at half the distance from the axle. 

If this principal axis be that of greatest moment of inertia, we must raise 
the screw I in order to bring it nearer the axle A. If it be the axis of least 
moment we must lower the screw /. In this way we may make the principal 
axis coincide with the axle. Let us suppose that the principal axis is that of 
greatest moment of inertia, and that we have made it coincide with the axle of 
the instrument. Let us also suppose that the moments of inertia about the 
other axes are equal, and very little less than that about the axle. Let the top 
be spun about the axle and then receive a disturbance which causes it to spin 
about some other axis. The instantaneous axis wiU not remain at rest either 
in space or in the body. In space it will describe a right cone, completing a 
revolution in somewhat less than the time of revolution of the top. In the 
body it will describe another cone of larger angle in a period which is longer 
as the difierence of axes of the body is smaller.' The invariable axis will be 
fixed in space, and describe a cone in the body. 

The relation of the different motions may be understood from the following 
illustration. Take a hoop and make it revolve about a stick which remains at 
rest and touches the inside of the hoop. The section of the stick represents the 



258 ON A DYNAinCAL TOP. 

path of the instantaneous axis in space, the hoop that of the same axis in the 
body, and the axis of the stick the invariable axis. The point of contact repre- 
sents the pole of the instantaneous axis itself, travelling many times round the 
stick before it gets once round the hoop. It is easy to see that the direction in 
which the instantaneous axis travels round the hoop, is in this case the same as 
that in which the hoop moves round the stick, so that if the top be spinning in 
the direction i, M, N, the colours will appear in the same order. 

By screwing the bob B up the axle, the difference of the axes of inertia 
may be diminished, and the time of a complete revolution of the invariable 
axis in the body increased. By observing the number of revolutions of the top 
in a complete cycle of colours of the invariable axis, we may determine the 
ratio of the moments of inertia. 

By screwing the bob up farther, we may make the axle the principal axis of 
least moment of inertia. 

The motion of the instantaneous axis will then be that of the point of 
contact of the stick with the outside of the hoop rolling on it. The order of 
colours will be N, M, L, if the top be spinning in the direction Z, M, N, and 
the more the bob is screwed up, the more rapidly will the colours change, till 
it ceases to be possible to make the observations correctly. 

In calculating the dimensions of the parts of the instrument, it is necessary 
to provide for the exhibition of the instrument with its axle either the greatest 
or the least axis of inertia. The dimensions and weights of the parts of the top 
which I have found most suitable, are given in a note at the end of this paper. 

Now let us make the axes of inertia in the plane of the ring unequal. We 
may do this by screwing the balance screws x and x^ farther from the axle 
without altering the centre of gravity. 

Let us suppose the bob B screwed up so as to make the axle the axis of 
least inertia. Then the mean axis is parallel to xt^, and the greatest is at right 
angles to xdd^ in the horizontal plane. The path of the invariable axis on the 
disc is no longer a circle but an ellipse, concentric with the disc, and having 
its major axis parallel to the mean axis xo^. 

The smaller the difference between the moment of inertia about the axle and 
about the mean axis, the more eccentric the ellipse will be; and if, by screwing 
the bob down, the axle be made the mean axis, the path of the invariable axis 
will be no longer a closed curve, but an hyperbola, so that it will depart alto- 
gether from the neighbourhood of the axle. When the top is in this condition 



ON A DYNAMICAL TOP. 259 

it must be spun gently, for it is very difficult to manage it when its motion 
gets more and more eccentric. 

When the bob is screwed still farther down, the axle becomes the axis of 
greatest inertia, and a:x^ the least. The major axis of the ellipse described by 
the invariable axis will now be perpendicular to ccx", and the farther the bob 
is screwed down, the eccentricity of the ellipse will diminish, and the velocity 
with which it is described will increase. 

I have now described all the phenomena presented by a body revolving freely 
on its centre of gravity. If we wish to trace the motion of the invariable axis 
by means of the coloured sectors, we must make its motion very slow compared 
■vvith that of the top. It is necessary, therefore, to make the moments of inertia 
about the principal axes very nearly equal, and in this case a very small change 
in the position of any part of the top will greatly derange the 'position of the 
principal axis. So that when the top is well adjusted, a single turn of one of 
the screws of the ring is sufficient to make the axle no longer a principal axis, 
and to set the true axis at a considerable inclination to the axle of the top. 

All the adjustments must therefore be most carefully arranged, or we may 
have the whole apparatus deranged by some eccentricity of spinning. The method 
of making the principal axis coincide with the axle must be studied and prac- 
tised, or the first attempt at spinning rapidly may end in the destruction of 
the top, if not of the table on which it is spun. 



On the Earth's Motion. 

We must remember that these motions of a body about its centre of gra- 
vity, are not illustrations of the theory of the precession of the Equinoxes. 
Precession can be illustrated by the apparatus, but we must arrange it so that 
the force of gravity acts the part of the attraction of the sun and moon in 
producing a force tending to alter the axis of rotation. This is easily done by 
bringing the centre of gravity of the whole a little below the point on which 
it spins. The theory of such motions is far more easily comprehended than 
that which we have been investigating. 

But the earth is a body whose principal axes are unequal, and from the 
phenomena of precession we can determine the ratio of the polar and equatorial 
axes of the "central ellipsoid;" and supposing the earth to have been set in 
motion about any axis except the principal axis, or to have had its original 



260 ON A DYNAMICAL TOP. 

axis disturbed in any way, its subsequent motion would be that of the top 
when the bob is a little below the critical position. 

The axis of angular momentum would have an invariable position in space, 
and would travel with respect to the earth round the axis of figure with a velo- 

C—A 

city = 0) -— : — where w is the sidereal angular velocity of the earth. The apparent 

pole of the earth would travel (with respect to the earth) from west to east 

A 

round the true pole, completing its circuit in jy — ^ sidereal days, which appears 

to be about 325*6 solar days. 

The instantaneous axis would revolve about this axis in space in about 
a day, and would always be in a plane with the true axis of the earth and 
the axis of angular momentum. The effect of such a motion on the apparent 
position of a star would be, that its zenith distance would be increased and 
diminished during a period of 325-6 days. This alteration of zenith distance 
is the same above and below the pole, so that the polar distance of the star 
is unaltered. In fact the method of finding the pole of the heavens by obser- 
vations of stars, gives the pole of the invan-aUe axis, which is altered only by 
external forces, such as those of the sun and moon. 

There is therefore no change in the apparent polar distance of stars due to 
this cause. It is the latitude which varies. The magnitude of this variation 
cannot be determined by theory. The periodic time of the variation may be 
found approximately from the known dynamical properties of the earth. The 
epoch of maximum latitude cannot be found except by observation, but it must 
be later in proportion to the east longitude of the observatory. 

In order to determine the existence of such a variation of latitude, I have 
examined the observations of Polaris with the Greenwich Transit Circle in the 
years 1851-2-3-4. The observations of the upper transit during each month were 
collected, and the mean of each month found. The same was done for the lower 
transits. The difference of zenith distance of upper and lower transit is twice 
the polar distance of Polaris, and half the sum gives the co-latitude of Greenwich. 

In this way I found the apparent co-latitude of Greenwich for each month 
of the four years specified. 

There appeared a very slight indication of a maximum belonging to the set 
of months, 

March, 51. Feb. 52. Dec. 52. Nov. 53. Sept. 54. 



ON A DYNAMICAL TOP. 261 

Tliis result, liowever, is to be regarded as very doubtful, as there did not 
appear to be evidence for any variation exceeding half a second of space, and 
more observations would be required to establish the existence of so small a 
variation at all. 

I therefore conclude that the earth has been for a long time revolving 
about an axis very near to the axis of figure, if not coinciding with it. The 
cause of this near coincidence is either the original softness of the earth, or 
the present fluidity of its interior. The axes of the earth are so nearly equal, 
that a considerable elevation of a tract of country might produce a deviation 
of the principal axis within the limits of observation, and the only cause which 
would restore the uniform motion, would be the action of a fluid which would 
gradually diminish the oscillations of latitude. The permanence of latitude essen- 
tially depends on the inequality of the earth's axes, for if they had been all 
equal, any alteration of the crust of the earth would have produced new prin- 
cipal axes, and the axis of rotation would travel about those axes, altering the 
latitudes of all places, and yet not in the least altering the position of the 
axis of rotation among the stars. 

Perhaps by a more extensive search and analysis of the observations of 
different observatories, the nature of the periodic variation of latitude, if it exist, 
may be determined. I am not aware of any calculations having been made to 
prove its non-existence, although, on dynamical grounds, we have every reason 
to look for some very small variation having the periodic time of 325-6 days 
nearly, a period which is clearly distinguished from any other astronomical cycle, 
and therefore easily recognised. 



262 ON A DYNAMICAL TOP. 

NOTK 

Dimensions and Weights of the parts of the Dynamical Top. 

I. Body of the top — 

Mean diameter of ring, 4 inches. 
Section of ring, | inch square. 

The conical portion rises from the upper and inner edge of the ring, a 
height of 1| inches from the base. 

The whole body of the top weighs 1 lb. 7 oz. 

Each of the nine adjusting screws has its screw 1 inch long, and the 

screw and head together weigh 1 ounce. The whole weigh . . 9 „ 

II. Axle, &c.— 

Length of axle 5 inches, of which | inch at the bottom is occupied by 
the steel point, 3J inches are brass with a good screw turned on it, 
and the remaining inch is of steel, with a sharp point at the top. 

The whole weighs 1^ „ 

The bob B has a diameter of 1'4< inches, and a thickness of •4. It weighs 2| „ 

The nuts b and c, for clamping the bob and the body of the top on the 

axle, each weigh ^ oz. 1 „ 



Weight of whole top 2 lb. 5J oz. 

The best arrangement, for general observations, is to have the disc of card divided 
into four quadrants, coloured with vermilion, chrome yellow, emerald green, and ultramarine. 
These are bright colours, and, if the vermilion is good, they combine into a grayish tint 
when the revolution is about the axle, and burst into brilliant colours when the axis is 
disturbed. It is useful to have some concentric circles, drawn with ink, over the colours, 
and about 12 radii drawn in strong pencil lines. It is easy to distinguish the ink from 
the pencil lines, as they cross the invariable axis, by their want of lustre. In this way, 
the path of the invariable axis may be identified with great accuracy, and compared with 
theory. 



VOL. I . PLATE III. 



riG 1 



FIG. 2 




PIG 4- 



VOL. I . PLATE III 




riG 6 



[From the Philosophical Magazine, Vol. xiv.] 

XVI. Account of Experiments on the Perception of Colour. 

To the Editors of the Philosophical Magazine and Journal. 

Gentlemen, 

The experiments which I intend to describe were undertaken in order 
to render more perfect the quantitative proof of the theory of three primary 
colours. According to that theory, every sensation of colour in a perfect human 
eye is distinguished by three, and only three, elementary qualities, so that in 
mathematical language the quahty of a colour may be expressed as a function 
of three independent variables. There is very little evidence at present for 
deciding the precise tints of the true primaries. I have ascertained that a 
certain red is the sensation wanting in colour-blind eyes, but the mathematical 
theory relates to the number, not to the nature of the primaries. If, with Sir 
David Brewster, we assume red, blue, and yellow to be the primary colours, this 
amounts to saying that every conceivable tint may be produced by adding 
together so much red, so much yellow, and so much blue. This is perhaps the 
best method of forming a provisional notion of the theory. It is evident that if 
any colour could be found which could not be accurately defined as so much of 
each of the three primaries, the theory would fall to the ground. Besides this, 
the truth of the theory requires that every mathematical consequence of assu m i n g 
every colour to be the result of mixture of three primaries should also be true. 

I have made experiments on upwards of 100 diiferent artificial colours, con- 
sisting of the pigments used in the arts, and their mechanical mixtures. These 
experiments were made primarily to trace the effects of mechanical mixture on 
various coloured powders ; but they also afford evidence of the truth of the 
theory, that all these various colours can be referred to three primaries. The 



264 EXPEKIMENTS ON THE PERCEPTION OF COLOUR. 

following experiments relate to the combinations of six well-defined colours only, 
and I shall describe them the more minutely, as I hope to induce those who 
have good eyes to subject them to the same trial of skill in distinguishing 
tints. 

The method of performing the experiments is described in the Transactions 
of the Royal Society of Edinburgh, Vol. xxi. Part 2. The colour- top or teetotum 
which I used may be had of Mr J. M. Bryson, Edinburgh, or it may be easily 
extemporized. Any rotatory apparatus which will keep a disc revolving steadily 
and rapidly in a good light, without noise or disturbance, and can be easily 
stopped and shifted, will do as well as the contrivance of the spinning-top. 

The essential part of the experiment consists in placing several discs of 
coloured paper of the same size, and slit along a radius, over one another, so 
that a portion of each is seen, the rest being covered by the other discs. By 
sliding the discs over each other the proportion of each colour may be varied, 
and by means of divisions on a circle on which the discs lie, the proportion of 
each colour may be read off. My circle was divided into 100 parts. 

On the top of this set of discs is placed a smaller set of concentric discs, 
so that when the whole is in motion round the centre, the colour resulting from 
the mixture of colours of the small discs is seen in the middle of that arising 
from the laro-er discs. It is the object of the experimenter to shift the colours 
till the outer and inner tints appear exactly the same, and then to read off the 
proportions. 

It is easy to deduce from the theory of three primary colours what must 
be the number of discs exposed at one time, and how much of each colour must 
appear. 

Every colour placed on either circle consists of a certain proportion of each 
of the primaries, and in order that the outer and inner circles may have precisely 
the same resultant colour in every respect, there must be the same amount of 
each of the primary colours in the outer and inner circles. Thus we have as 
many conditions to fulfil as there are primary colours; and besides these we 
have two more, because the whole number of divisions in either the outer or 
the inner circle is 100, so that if there are three primary colours there wiU be 
five conditions to fulfil, and this will require five discs to be disposable, and 
these must be arranged so that three are matched against two, or four against one. 
If we take six difierent colours, we may leave out any one of the six, and 
so form six different combinations of five colours. It is plain that these six 



EXPERIMENTS ON THE PERCEPTION OF COLOUR. 265 

combinations must be equivalent to two equations only, if the theory of three 
primaries be true. 

The method which I have found most convenient for registering the result 
of an experiment, after an identity of tint has been obtained in the inner and 
outer circles, is the following : — 

Write down the names or symbols of the coloured discs each at the top of 
a column, and underneath write the number of degrees of that colour observed, 
calling it + when the colour is in the outer circle, and — when it is in the inner 
circle ; then equate the whole to zero. In this way the account of each colour 
is kept in a separate column, and the equations obtained are easily combined and 
reduced, without danger of confounding the colours of which the quantities have 
been measured. The following experiments were made between the 3rd and 11th 
of September, 1856, about noon of each day, in a room fronting the north, 
without curtains or any bright- coloured object near the window. The same 
combination was never made twice in one day, and no thought was bestowed 
upon the experiments except at the time of observation. Of course the gradua- 
tion was never consulted, nor former experiments referred to, till each combi- 
nation of colours had been fixed by the eye alone; and no reduction waa 
attempted till all the experiments were concluded. 

The coloured discs were cut from paper painted of the following colours : — 
Vermilion, Ultramarine, Emerald-green, Snow-white, Ivory-black, and Pale 
Chrome-yellow. They are denoted by the letters V, U, G, W, B, Y respectively. 
These colours were chosen, because each is well distinguished from the rest, so 
that a small change of its intensity in any combination can be observed. Two 
discs of each colour were prepared, so that in each combination the colours might 
occasionally be transposed from the outer circle to the inner. 

The first equation was formed by leaving out vermilion. The remaining 
colours are Ultramarine-blue, Emerald-green, White, Black, and Yellow. We 
might suppose, that by mixing the blue and yellow in proper proportions, we 
should get a green of the same hue as the emerald-green, but not so intense, 
80 that in order to match it we should have to mix the green with white to 
dilute it, and with black to make it darker. But it is not in this way that we 
have to arrange the colours, for our blue and yellow produce a pinkish tint, and 
never a green, so that we must add green to the combination of blue and yellow, 
to produce a neutral tint, identical with a mixture of white and black. 



266 EXPERIMENTS ON THE PERCEPTION OF COLOUR. 

Blue, green, and yellow must therefore be combined on the large discs, and 
stand on one side of the equation, and black and white, on the small discs, must 
stand on the other side. In order to facilitate calculations, the colours are 
always put down in the same order; but those belonging to the small discs 
are marked negative. Thus, instead of writing 

54U + UG + 32Y = 32W + 68B, 
we write +54U + 14G-32W-68B + 32Y = 0. 

The sum of all the positive terms of such an equation is 100, being the 
whole number of divisions in tne circle. The sum of the negative terms is 
also 100. 

The second equation consists of all the colours except blue ; and in this 
way we obtain six different combinations of five colours. 

Each of these combinations was formed by the unassisted judgment of my 
eye, on six different occasions, so that there are thirty-six independent observa- 
tions of equations between five colours. 

Table I. gives the actual observations, with their dates. 

Table II. gives the result of summing together each group of six equations. 

Each equation in Table 11. has the sums of its positive and negative co- 
eflBcients each equal to 600. 

Having obtained a number of observations of each combination of colours, 
we have next to test the consistency of these results, since theoretically two 
equations are sufficient to determine all the relations among six colours. We 
must therefore, in the first place, determine the comparative accuracy of the 
different sets of observations. Table III. gives the averages of the errors of 
each of the six groups of observations. It appears that the combination IV. is 
the least accurately observed, and that VI. is the best. 

Table IV. gives the averages of the errors in the observation of each colour 
in the whole series of experiments. This Table was computed in order to detect 
any tendency to colour-blindness in my own eyes, which might be less accurate 
in discriminating red and green, than in detecting variations of other colours. 
It appears, however, that my observations of red and green were more accurate 
than those of blue or yellow. White is the most easily observed, from the 



EXPERIMENTS ON THE PERCEPTION OF COLOUR. 267 

brilliancy of the colour, and black is liable to the greatest mistakes. I would 
recommend this method of examining a series of experiments as a means of 
detecting partial colour-blindness, by the different accuracy in observing differ- 
ent colours. The next operation is to combine all the equations according to 
their values. Each was first multiplied by a coefficient proportional to its ac- 
curacy, and to the coefficient of white in that equation. The result of adding 
all the equations so found is given in equation (W). 

Equation (Y) is the result of similar operations with reference to the 
yellow on each equation. 

We have now two equations, from which to deduce six new equations, by 
eliminating each of the six colours in succession. We must first combine the 
equations, so as to get rid of one of the colours, and then we must divide by 
the sum of the positive or negative coefficients, so as to reduce the equations 
to the form of the observed equations. The results of these operations are given 
in Table V., along with the means of each group of six observations. It will 
be seen that the differences between the results of calculation from two equations 
and the six independent observed equations are very small. The errors in red 
and green are here again somewhat less than in blue and yellow, so that there 
is certainly no tendency to mistake red and green more than other colours. 
The average difference between the observed mean value of a colour and the 
calculated value is 77 of a degree. The average error of an observation in any 
group from the mean of that group was '92. No observation was attempted 
to be registered nearer than one degree of the top, or yo7 of ^ circle ; so that 
this set of observations agrees with the theory of three primary colours quite 
as far as the observations can warrant us in our calculations ; and I think that 
the human eye has seldom been subjected to so severe a test of its power of 
distinguishing colours. My eyes are by no means so accurate in this respect as 
many eyes I have examined, but a little practice produces great improvement 
even in inaccurate observers. 

I have laid down, according to Newton's method, the relative positions of 
the five positive colours with which I worked. It will be seen that W lies 
within the triangle VUG, and Y outside that triangle. 

The first combination. Equation I., consisted of blue, yellow, and green, 
taken in such proportions that their centre of gravity falls at W, 



268 



EXPERIMENTS ON THE PERCEPTION OF COLOUR. 



In Equation II. a mixture of red and green, represented in the diagram 
by the point 2, is seen to be equivalent to a mixture of white and yellow, also 
represented by 2, which is a pale yellow tint. 

Equation III. is between a mixture of blue and yellow and another of 
white and red. The resulting tint is at the intersection of YU and WV ; that 
is, at the point 3, which represents a pale pink grey. 

Equation IV. is between VG and UY, that is, at 4, a dirty yellow. 

Equation V. is between a mixture of white, red, and green, and a mixture 
of blue and yellow at the point 5, a pale dirty yellow. 

Equation VI. has W. for its resulting tint. 



Blue, U. 



Bed, V 




G, Green. 



Y, Yellow. 

Of all the resulting tints, that of Equation IV. is the furthest from white ; 
and we find that the observations of this equation are affected with the greatest 
errors. Hence the importance of reducing the resultant tint to as nearly a 
neutral colour as possible. 

It is hardly necessary for me to observe, that the whole of the numerical 
results which I have given apply only to the coloured papers which I used, 
and to them only when illuminated by daylight from the north at mid-day in 
September, latitude 55". In the evening, or in winter, or by candlelight, the 
results are very different. I believe, however, that the results would differ far 
less if observed by different persons, than if observed under different lights ; 
for the apparatus of vision is wonderfully similar in different eyes, and even in 
colour-blind eyes the system of perception is not different, but defective. 



EXPERIMENTS ON THE PERCEPTION OF COLOUR. 



269 



Table I. — The observations arranged in groups. 



Equation I. 


V = 0. 


+ U. 


+ G. 


-W. 


-B. +Y. 


Equation IV. 


-V. 


+u. 


-O. 


w=o. 


+ B. 


+ Y. 


1856, Sept. 3. 





54 


12 


34 


66 34 


1856, 


Sept. 3. 


62 


15 


38 





53 


32 


4. 





58 


14 


31 


69 28 




4. 


63 


17 


37 





46 


37 


5. 





55 


12 


32 


68 33 




5. 


64 


16 


36 





50 


34 


6. 





54 


14 


32 


68 32 




6. 


62 


19 


38 





46 


35 


8. 





54 


14 


32 


68 32 




8. 


62 


19 


38 





47 


34 


9. 





53 


15 


32 


68 32 




9. 


63 


17 


37 





49 


34 


Equation n. 


-V. 


u=o. 


-G. 


+ \V. 


+ B. +Y. 


Equation V. 


+v. 


-U. 


+ G. 


+w. 


B = 0. 


-Y. 


Sept. 3. 


59 





41 


9 


71 20 




Sept. 3. 


56 


47 


28 


16 





53 


4. 


61 





39 


9 


68 23 




4. 


57 


50 


25 


18 





50 


5. 


61 





39 


9 


67 24 




5. 


66 


49 


24 


20 





51 


6. 


59 





41 


10 


66 24 




6. 


55 


47 


27 


18 





53 


8. 


60 





40 


9 


69 22 




8. 


54 


49 


26 


20 





51 


9. 


61 





39 


9 


68 23 




11. 


56 


50 


27 


17 





50 


Equation HI. 


+v. 


-u. 


G = 0. 


+w. 


+ B. -Y. 


Equation VI. 


+v. 


+ U. 


+ G. 


-W. 


-B. 


Y = 0. 


Sept. 3. 


20 


56 





28 


52 44 




Sept. 3. 


38 


27 


35 


24 


76 





4. 


23 


58 





30 


47 42 




4. 


39 


27 


34 


24 


76 





5. 


24 


56 





29 


47 44 




5. 


40 


26 


34 


24 


76 





6. 


20 


56 





31 


49 44 




6. 


38 


28 


34 


24 


76 





8. 


21 


57 





29 


60 43 




8. 


39 


28 


33 


24 


76 





9. 


21 


58 





29 


50 42 




11. 


39 


27 


34 


23 


77 









Table II.— The sums of the observed equations. 


















V. 


U. 


G. 


W. 




B. 


Y. 








Equation I. 







+ 328 


+ 81 


-193 


- 


-407 


+ 


191 










II. 


_ 


361 





-239 


+ 55 


+ 409 


+ 


136 










III. 


+ 


129 


-341 





+ 176 


+ 295 


_ 


259 










IV. 




376 


+ 103 


-224 





+ 291 


+ 


206 










V. 


+ 


334 


-292 


+ 157 


+ 109 







- 


308 










VI. 


4 


233 


+ 163 


+ 204 


-143 


- 


-457 












Table III. — The averages of the errors of the several equations from the means expressed in 

j^ parts of a circle. 



Equations. 


I. 


n. 


m. 


IV. 


V. 


VL 


Errors. 


•94 


•85 


1-05 


117 


ro8 


•40 



Table IV. — The averages of the errors of the several colours from the means in y^ parts of 

a circle. 
Colours. V. D. G. W. B, Y. 

Errors. -83 -99 •SO -61 115 r09 

Average error on the whole ^92. 

The equations from which the reduced results were obtained were calculated as follow : — 
Equation for (W)- (II) + 2 (III) + (V)-2 (I) -4 (VI). 
Equation for (Y) = 2 (I) + 2 (II) - 3 (III) + 2 (IV) - 3 (V> 



270 EXPERIMENTS ON THE PERCEPTION OF COLOUR. 

These operations being performed, gave 

V. U. G. W. B. Y. 

(W) + 701 + 2282 + 1060-1474-3641 + 1072 = 0. 
(Y) +2863-2761 + 1235 + 1131^ 299-2767 = 0. 



From these were obtained the following results by elimination: — 

Table V. 



Equation 

J r From (W) and (Y) 
■ \ From observation 






-54-1 
-54-7 


-13-9 
-13-5 


+ 32-0 
+ 32-1 


+ 68-0 
+ 67-9 


-32 
-31-8 


jj ( From (W) and (Y) 
* ( From observation 


-59-6 
-60-2 






-40-4 
-39-8 


+ 10-4 
+ 9-2 


+ 66-0 
+ 68-2 


+ 23-6 
+ 22-6 


,^^ f From (W) and (Y) 
\ From observation 


-21-7 
-21-5 


+ 57-4 
+ 56-8 






-30-2 
-29-3 


-48-1 
-49-2 


+ 42-6 
+ 43-2 


f From (W) and (Y) 
( From observation 


-62-4 
-62-7 


+ 18-6 
+ 17-2 


-37-6 
-37-3 






+ 45-7 
+ 48-5 


+ 35-7 
+ 34-3 


1 From (W) and (Y) 
■ ( From observation 


+ 55-6 
+ 55-7 


-49-0 

-48-7 


+ 25-2 
+ 26-1 


+ 19-2 
+ 18-2 






-51-0 
-51-3 



^T f From (W) and (Y) -397 -26-6 -337 +227 +77-3 

^^•\ From observation -38-8 -27-2 -340 +28-3 +76-2 

James Clerk Maxwell. 
Glexlair, Jum 13, 1857. 



[From The Quarterly Journal of Pure and Applied Mathematics, Vol. ii. 



XVII. On the General Laws of Optical Instruments. 

The optical effects of compound instruments have been generally deduced 
from those of the elementary parts of which they are composed. The formulae 
given in most works on Optics for calculating the effect of each spherical sur- 
face are simple enough, but, when we attempt to carry on our calculations from 
one of these surfaces to the next, we arrive at fractional expressions so com- 
phcated as to make the subsequent steps very troublesome. 

Euler (Acad. R. de Berlin, 1757, 1761. Acad. R. de Paris, 1765) has attacked 
these expressions, but his investigations are not easy reading. Lagrange (Acad. 
Berhn, 1778, 1803) has reduced the case to the theory of continued fractions 
and so obtained general laws. 

Gauss [Dioptrische Untersuchungen, Gottingen, 1841) has treated the subject 
with that combination of analytical skiU with practical ability which he displays 
elsewhere, and has made use of the properties of principal foci and principal 
planes. An account of these researches is given by Prof. Miller in the third 
volume of Taylor's Scientific Memoirs. It is also given entire in French by 
M. Bravais in Liouvilles Journal for 1856, with additions by the translator. 

The method of Gauss has been followed by Prof Listing in his Treatise 
on the DioptHcs of the Eye (in Wagner's Handworterhuch der Physiologie) from 
whom I copy these references, and by Prof Helmholtz in his Treatise vn 
Physiological Optics (in Karsten's Cyclopadie). 

The earliest general investigations are those of Cotes, given in Smith's 
Optics, II. 76 (1738). The method there is geometrical, and perfectly general, 
but proceeding from the elementary cases to the more complex by the method 
of mathematical induction. Some of his modes of expression, as for instance his 
measure of "apparent distance," have never come into use, although his results 
may easily be expressed more intelligibly ; and indeed the whole fabric of 



272 ON THE GENERAL LAWS OF OPTICAL INSTRUMENTS. 

Geometrical Optics, as conceived by Cotes and laboured by Smith, has fallen 
into neglect, except among the writers before named. Smith tells us that it 
was with reference to these optical theorems that Newton said " If Mr Cotes 
had lived we might have known something." 

The investigations which I now offer are intended to show how simple and 
how general the theory of instruments may be rendered, by considering the 
optical effects of the entire instrument, without examining the mechanism by 
which those effects are obtained. I have thus established a theory of "perfect 
instruments," geometrically complete in itself, although I have also shown, that 
no instrument depending on refraction and reflexion, (except the plane mirror) 
can be optically perfect. The first part of this theory was conununicated to 
the Philosophical Society of Cambridge, 28th April, 1856, and an abstract will 
be found in the Philosophical Magazine, November, 1856. Propositions VIII. 
and IX. are now added. I am not aware that the last has been proved before. 

In the following propositions I propose to establish certain rules for deter- 
mining, from simple data, the path of a ray of light after passing through any 
optical instrument, the position of the conjugate focus of a luminous point, and 
the magnitude of the image of a given object. The method which I shall use 
does not require a knowledge of the internal construction of the instrument and 
derives all its data from two simple experiments. 

There are certain defects incident to optical instruments from which, in the 
elementary theory, we suppose them to be free. A perfect instrument must 
fulfil three conditions : 

I. Every ray of the pencil, proceeding from a single point of the object, 
must, after passing through the instrument, converge to, or diverge from, a 
single point of the image. The corresponding defect, when the emergent rays 
have not a common focus, has been appropriately called (by Dr Whewell) 
Astigmatism. 

II. If the object is a plane surface, perpendicular to the axis of the 
instrument, the image of any point of it must also lie in a plane perpendicular 
to the axis. When the points of the image lie in a curved surface, it is said 
to have the defect of curvature. 

III. The image of an object on this plane must be similar to the object, 
whether its linear dimensions be altered or not; when the image is not similar 
to the object, it is said to be distorted. 



ox THE GENERAL LAWS OF OPTICAL INSTRUMENTS. 273 

An image free from these three defects is said to be jycrfect. 

In Fig. 1, p. 285, let A^x^a^ represent a plane object perpendicular to the 
axis of an instrument represented by I., then if the instrument is perfect, as 
regards an object at that distance, an image A.a.p^_ will be formed by the 
emergent rays, which will have the following properties : 

I. Every ray, which passes through a point a^ of the object, will pass 
through the corresponding point a. of the image. 

II. Every point of the image will lie in a plane perpendicular to the axis. 

III. The figure A.ap^ will be similar and similarly situated to the figure 

Now let us assume that the instrument is also perfect as regards an object 

in the plane i?i?>,y8i perpendicular to the axis through -B„ and that the image 

of such an object is in the plane B^fio and similar to the object, and we 
shall be able to prove the following proposition : 

Prop. I. If an instrument give a perfect image of a plane object at two 
different distances from the instrument, all incident rays having a common focus 
will have a common focus after emergence. 

Let Pj be the focus of incident rays. Let P-,a^^ be any incident ray. 
Then, since every ray which passes through a^ passes through a,,, its image after 
emergence, and since every ray which passes through Z;, passes through 6,, the 
direction of the ray P^a^\ after emergence must be ah.. 

Similarly, since a^ and ySj are the images of Oj and ^i, if P^a^^^ be any 
other ray, its direction after emergence will be a„fi.y 

Join a, a,, h^^„ a.xL.., hfi.,; then, since the parallel planes AjCt^a^ and BJ}^, 
are cut by the plane of the two rays through P^, the intersections cTiOi and 
?jjSi are parallel. 

Also, their images, being similarly situated, are parallel to them, therefore 
a„a, is parallel to 6^j, and the lines aJj„ and a,^^ are in the same plane, and 
therefore either meet in a point P^ or are parallel. 

Now take a third ray through P,, not in the plane of the two former. 
After emergence it must either cut both, or be parallel to them. If it cuts 
both it nuist pass through the point P., and then every other ray must pass 
through P., for no line can intersect three Hues, not in one plane, without 
passing through their point of intersection. If not, then all the emergent rays 



274 ON THE GENERAL LAWS OF OPTICAL INSTRUMENTS. 

are parallel, which is a particular case of a perfect pencil. So that for every 
position of the focus of incident rays, the emergent pencil is free from astig- 
matism. 

Prop. II. In an instrument, perfect at two different distances, the image 
of any plane object perpendicular to the axis will be free from the defects of 
curvature and distortion. 

Through the point P, of the object draw any line P,Q, in the plane of 
the object, and through P,Q, draw a plane cutting the planes A„ B, in the hnes 
ttio,, h^,. These lines will be parallel to P,Q, and to each other, wherefore 
also their images, a^o,, b^„ will be parallel to P,Q, and to each other, and 
therefore in one plane. 

Now suppose another plane drawn through P^Q, cutting the planes A, and 
B, in two other lines parallel to P,Q^. These will have parallel images in the 
planes A^ and B„ and the intersection of the planes passing through the two 
pairs of images wiU define the line P^Q, which will be parallel to them, and 
therefore to P,Q„ and will be the image of P,Q,. Therefore P^, the image 
of P,Qi is parallel to it, and therefore in a plane perpendicular to the axis. 
Now if all corresponding lines in any two figures be parallel, however the lines 
be drawn, the figures are similar, and similarly situated. 

From these two propositions it follows that an instrument giving a perfect 
image at two different distances will give a perfect image at all distances. We 
have now only to determine the simplest method of finding the position and 
magnitude of the image, remembering that wherever two rays of a pencil inter- 
sect, all other rays of the pencil must meet, and that aU parts of a plane 
object have their images in the same plane, and equally magnified or diminished. 

Prop. III. A ray is incident on a perfect instrument parallel to the axis, 
to find its direction after emergence. 

Let a J), (fig. 2) be the incident ray, A,a, one of the planes at which an 
object has been ascertained to have a perfect image. A,a, that image, similar 
to A^tti but in magnitude such that A/t^^xA.a,. 

Similarly let BJ), be the image of BJj„ and let BM, = yBA- Also let 
A,B, = c, and A.X^ = c^. 

Then since a, and h„ are the images of a, and \, the line F^aK will be 
the direction of the ray after emergence, cutting the axis in F^, (unless x = y. 



ON THE GENERAL LAWS OF OPTICAL INSTRUMENTS. 275 

when a.})^ becomes parallel to the axis). The point F._ may be found, by 
remembering that A^a, = B^b^, Ajii = xAfL^, B]j. = yDJj^. We find — 

■ " 'y-x 

Let g^ be the point at which the emergent ray is at the same distance 
from the axis as the incident ray, draw gfi^ perpendicular to the axis, then 
we have 

' y-x 

Similarly, if aSiF^ be a ray, which, after emergence, becomes parallel to 
the axis ; and gfi^ a line perpendicular to the axis, equal to the distance of 
the parallel emergent ray, then 

A,F, = c,-y~, F,G,^^^^ . 
x—y ^—y 

Definitions. 

I. The point F^, the focus of incident rays when the emergent rays are 
parallel to the axis, is called the Jirst jprincii^al focus of the instrument. 

II. The plane G^^ at which incident rays through F^ are at the same 
distance from the axis as they are after emergence, is called the first princi- 
pal plane of the instrument. F^G^ is called the first focal length. 

III. The point F^, the focus of emergent rays when the incident rays 
are parallel, is called the second principal focus. 

IV. The plane G,^., at which the emergent rays are at the same distance 
from the axis, as before incidence, is called the second principal plane, and 
Ffi^ is called the second focal length. 

When x = y, the ray is parallel to the axis, both at incidence and emerg- 
ence, and there are no such points as F and G. The instrument is then 
called a telescope. x( = y) is called the linear ina^nifying power and is denoted 

by I, and the ratio - is denoted by n, and may be called the elongation. 

In the more general case, in which x and y are different, the principal 
foci and principal planes afford the readiest means of finding the position of 
images. 



276 ON THE GENERAL LAWS OF OPTICAL INSTRUMENTS. 

Prop. IV. Given the principal foci and principal planes of an instrument, 
to find the relations of the foci of the incident and emergent pencils. 

Let F„ F„ (fig. 3) be the principal foci, G^, G., the principal planes, Q^ 
the focus of incident light, Q^P^ perpendicular to the axis. 

Through ^1 draw the ray Q^g^F^. Since this ray passes through F^ it 
emerges parallel to the axis, and at a distance from it equal to G^g^. Its 
direction after emergence is therefore Q.,g^ where G^g„ = G^g^. Through Q^ draw 
Q{Yi parallel to the axis. The corresponding emergent ray wiD pass through 
F^^, and will cut the second principal plane at a distance G^y^_= G-^y^, so that 
jP„y, is the direction of this ray after emergence. 

Since both rays pass through the focus of the emergent pencil, Q^, the 
point of intersection, is that focus. Draw Q^P^ perpendicular to the axis. 
Then PxQi = G{Y^ = G^y., and G,g, = G^g^ = P,Q.,. By similar triangles F,P,Q, and 

■F.G^r 

P,F, : F,G, :: P,Q, : {G,g, = ) P,Q,. 
And by similar triangles F^P^Q^ and F^G^y^ 

Pm = Gry^) ■ P^Q^ ■■■■ ^^. ■■ F^P^-- 
We may put these relations into the concise form 

P,F,_P,Q,_G^, 

F^r p.Qr F,p,' 

and the values of F„P^ and PJ^^ are 

F G GJF F G 

F..P.= '^'pf^" - and P.Q. = ^'P.Q,. 

These expressions give the distance of the image from F^ measured along the 
axis, and also the perpendicular distance from the axis, so that they serve to 
determine completely the position of the image of any point, when the princi- 
pal foci and principal planes are known. 

Prop. V. To find the focus of emergent rays, when the instrument is a 
telescope. 

Let ^1 (fig. 4) be the focus of incident rays, and let Q^aJ)^ be a ray 
parallel to the axis ; then, since the instrument is telescopic, the emergent 
ray Q^aM^ will be parallel to the axis, and Q^P^^l. Q^P^. 



ON THE GENERAL LAWS OF OPTICAL INSTRUMENTS. 277 

Let QiOiB^ be a ray through ^,, the emergent ray will be Q,a,J5,, and 

AM, ~ A,a,~ I. A,a, " A.a, " A,B, ' 

so that -FT^ = -4 r>' = n, a constant ratio. 

P^B, A,B^ 

Cor. If a point C be taken on the axis of the instrument so that 

^^^ = A,B,-A^, ^'^' = T:^ ^^^" 

then CP, = n.CP,. 

Def. The point C is called the centre of the telescope. 

It appears, therefore, that the image of an object in a telescope has its 
dimensions perpendicular to the axis equal to I times the corresponding dimen- 
sions of the object, and the distance of any part from the plane through C 
equal to n times the distance of the corresponding part of the object. Of 
course all longitudinal distances among objects must be multipUed by n to 
obtain those of their images, and the tangent of the angular magnitude of an 

object as seen from a given point in the axis must be multipHed by - to 

obtain that of the image of the object as seen from the image of the given 

point. The quantity - is therefore called the angular magnifying power, and 

is denoted by m. 

Prop. VI. To find the principal foci and principal planes of a combina- 
tion of two instruments having a common axis. 

Let /, /' (fig. 5) be the two instruments, G^F^Ffi, the principal foci and 
planes of the first, G^F^F^G^ those of the second, V^<^^^S, those of the com- 
bination. Let the ray g^jJj'g^ pass through both instruments, and let it be 
parallel to the axis before entering the fii'st instrument. It will therefore pass 
through F„ the second principal focus of the first instrument, and through g. 
so that G^^ = (xi(7i. 

On emergence from the second instrument it will pass through ^^ the 
focus conjugate to F,, and through g^ in the second principal plane, so that 



278 ON THE GENERAL LAWS OF OPTICAL INSTRUMENTS. 

(r.'g' = G^g^. (f>i is by definition the second principal focus of the combination 
of instruments, and if T^y^ be the second principal plane, then r„y, = G^g^ 

We have now to find the positions of <f>, and Tj. 

By Prop. IV., we have 

^^^== — F:Fr~ • 

Or, tlie distance of the principal focus of the combination, from that of the 
second instrument, is equal to the product of the focal lengths of the second 
instrument, divided by the distance of the second principal focus of the first 
instrument from the first of the second. From this we get 

r"jp' jp'A ^"'^^ {FjF^ — F^G() 
Ctj i^j - -t^2 9a = jrpT , 

oi G,<f>, = jrp7 . 

Now, by the pairs of similar triangles ^G^g^, (jtV^y, and FJjr(g', F^G^^, 

T,<j>, _ r,y, ^ %, _ F„G, 

~g:4>. Gig. G:g( g;f,- 

Multiplying the two sides of the former equation respectively by the first and 
last of these equal quantities, we get 

, Gr^ , . GiF„' 

Or, the second focal distance of a combination is the product of the second 
focal lengths of its two components, divided by the distance of their consecutive 
principal foci. 

If we call the focal distances of the first instrument f^ and /,, those of 
the second // and //, and those of the combination J\, /j, and put FJF^=d, 
then the positions of the principal foci are found fi:om the values 

and the focal lengths of the combination from 

'~ d ' J'~ d ' 



ON THE GENERAL LAWS OF OPTICAL INSTRUMENTS. 279 

When d = 0, all these values become infinite, and the compound instruiaent 
becomes a telescope. 

Prop. VII. To find the linear magnifying power, the elongation, and the 
centre of the instrument, when the combination becomes a telescope. 

Here (fig. 6) the second principal focus of the first instrument coincides at J' 
with the first of the second. (In the figure, the focal distances of both instru- 
ments are taken in the opposite direction from that formerly assumed. They are 
therefore to be regarded as negative.) 

In the first place, F,' is conjugate to F^, for a pencil whose focus before 
incidence is F^ will be parallel to the axis between the instruments, and will 
converge to i^/ after emergence. 

Also if G^g^ be an object in the first principal plane, G,g„ will be its first 
image, equal to itself, and if Hh be its final image 

^^^- Gjr-~- f:^ 

Now the linear magnifying power is 7,- , and the elongation is .' . 
because F.' and H are the images of F.^ and G^ respectively ; therefore 

l=-4^ and n=££-. 
The angular magnifying power = in = -= — 4-7 • 
The centre of the telescope is at the point C, such that 

When n becomes 1 the telescope has no centre. The efiect of the Instruineni 
is then simply to alter the position of an object by a certain distance measured 
along the axis, as in the case of refraction through a plate of glass bounded bv 
parallel planes. In certain cases this constant distance itself disappears, as in 
the case of a combination of three convex lenses of which the focal lengths arr 



280 ON THE GENERAL LAWS OF OPTICAL INSTRUMENTS. 

4, 1, 4 and the distances 4 and 4. This combination simply inverts every object 
without altering its magnitude or distance along the axis. 

The preceding theory of perfect instruments is quite independent of the 
mode in which the course of the rays is changed within the instrument, as 
we are supposed to know only that the path of every ray is straight before 
it enters, and after it emerges from the instrument. We have now to con- 
sider, how far these results can be applied to actual instruments, in which 
the course of the rays is changed by reflexion or refraction. "We know that 
such instruments may be made so as to fulfil approximately the conditions of 
a perfect instrument, but that absolute perfection has not yet been obtained. 
Let us inquire whether any additional general law of optical instruments can 
be deduced from the laws of reflexion and refraction, and whether the imper- 
fection of instruments is necessary or removeable. 

The following theorem is a necessary consequence of the known laws of 
reflexion and refraction, whatever theory we adopt. 

If we multiply the length of the parts of a ray which are in diflerent 
media by the indices of refraction of those media, and call the sum of these 
products the reduced path of the ray, then : 

I. The extremities of all rays from a given origin, which have the same 
reduced path, lie in a surface normal to those rays. 

II. When a pencil of rays is brought to a focus, the reduced path from 
the origin to the focus is the same for every ray of the pencil. 

In the undulatory theory, the " reduced path " of a ray is the distance 
through which light would travel in space, during the time which the ray 
takes to traverse the various media, and the surface of equal " reduced paths " 
is the wave-surface. In extraordinary refraction the wave-surface is not always 
normal to the ray, but the other parts of the proposition are true in this and all 
other cases. 

From this general theorem in optics we may deduce the following propo- 
sitions, true for all instruments depending on refraction and reflexion. 

Prop. VIII. In any optical instrument depending on refraction or reflex- 
ion, if ajtti, />i^i (fig. 7) be two objects and a.a.^, h.fi^ their images, A^B^ the 
distance of the objects, AM. that of the images, ^i^ the index of refraction of 



ON THE GENERAL LAWS OF OPTICAL INSTRUMENTS. 281 

the medium in which the objects are, /a, that of the medium in which tlie 
images are, then 

«,a, X /^,y8, _ a,a, x h.fi., 
^' A A ~^' A,B., '' 

approximately, when the objects are small. 

Since a, is the image of a^, the reduced path of the ray a,6,a,, will be 
equal to that of a^^a„_, and the reduced paths of the rays a^/3,cu and a,/Aa, will 
be equal. 

Also because l)^^ and h.^„ are conjugate foci, the reduced paths of the 
rays b^ajj, and h^aj),, and of ^ia,,/8j and ^,a.,/3, will be equal. So that the 
reduced paths 

afi, + h,a^ = a^ySj + ^.a^ 

aJ3, + I3,0L, = tti^i + b.cL, 

feiOj + Oj^j = b^a^ + alt., 

these being still the reduced paths of the rays, that is, the length of each 
ray multiplied by the index of refraction of the medium. 

If the figure is symmetrical about the axis, we may write the equation 

Fi (aA - «i^i) = /^2 (aA - ci-A), 
where aJS^, &c. are now the ax^tual lengths of the rays so named. 



Now aA' = A,B;' + 1 (a,a, + b^.f, 



so that a^i — aj)^ = OiC^ x 6^8, , 

a.a, X 61)8, 



and ft, (a^ - aj),) = fi^ 



a A + aj)^ 



Similarly /x, (a^ - a,&,) = fi, ^^^^^j ^' 

So that the equation /x, ^ , "T' = /x^ — ^— — , , 



282 ON THE GENERAL LAWS OF OPTICAL INSTRUMENTS. 

is true accurately, and since when the objects are small, the denominators are 
nearly 2A,B^ and 2A^„ the proposition is proved approximately true. 

Using the expressions of Prop. III., this equation becomes 

1 xy 

Now by Prop. III., when x and y are different, the focal lengths /, and /, 

are 

. xy ^ 1 

^1 'x-y ^ y — ^ 

therefore -^ = -^ = - by the present theorem. 

So that in any instrument, not a telescope, the focal lengths are directly as 
the indices of refraction of the media to which they belong. If, as in most 
cases, these media are the same, then the two focal distances are eqiial 

When x = y, the instrument becomes a telescope, and we have, by Prop. V., 

l = x and n=-; and therefore by this theorem 

m n' 

We may find I experimentally by measuring the actual diameter of the 
image of a known near object, such as the aperture of the object glass. If be 
the diameter of the aperture and o that of the circle of light at the eye-hole 
(which is its image), then 

From this we find the elongation and the angular magnifying power 

n = ^'l\ and m = ^'y. 

When ix, = fi„ as in ordinary cases, m = y = -, which is Gauss' rule for deter- 
mining the magnifying power of a telescope. 



ON THE GENERAL LAWS OF OPTICAL INSTRUMENTS. 283 

Prop. IX. It is impossible, bj means of any combination of reflexions 
and refractions, to produce a perfect image of an object at two different distances, 
unless the instrument be a telescope, and 

l = n=-, m=l. 

It appears from the investigation of Prop. VIII. that the results there 
obtained, if true when the objects are very small, will be incorrect when the 
objects are large, unless 

ajSi + tti^i : a^^ + a,h :: A^B^ : A^^, 

and it is easy to prove that this cannot be, unless all the Hnes in the one figure 
are proportional to the corresponding lines in the other. 

In this way we might show that we cannot in general have an astigmatic, 
plane, undistorted image of a plane object. But we can prove that we cannot 
get perfectly focussed images of an object in two positions, even at the expense 
of curvature and distortion. 

We shall first prove that if two objects have perfect images, the reduced 
path of the ray joining any given points of the two objects is equal to that 
of the ray joining the corresponding points of the images. 

Let tto (fig. 8) be the perfect image of a^ and yS^ of /B^. Let 

Ajai = a^, BJ3, = b„ Ajx^ = a^, B.J3., = b., A^B^ = c^, A^^ = c^. 

Draw a^D^ parallel to the axis to meet the plane B^y and aJD, to the plane 
of A. 

Since everything is symmetrical about the axis of the instrument we shall 
have the angles D^Bfi^ = D.M.fi, = d, then in either figure, omitting the sufl&xes, 

= c' + a' + b'-2ahcose. 

It has been shown in Prop. VIII. that the difference of the reduced paths 
of the rays aj)^, afi^ in the object must be equal to the difference of the reduced 
paths of a^^j, a^^ in the image. Therefore, since we may assume any value for 6 

/^i J{(^x + &i' + Ci* - lajb, cos 6) - fi, J{a^ + h^ + c^ - 2a,h cos 6) 



284 ON THE GENERAL LAWS OF OPTICAL INSTRUMENTS. 

13 constant for all values of 6. This can be only when 

and fi, J{aJ),) =fi,J (aM,), 

which shows that the constant must vanish, and that the lengths of lines 
joining corresponding points of the objects and of the images must be inversely 
as the indices of refraction before incidence and after emergence. 

Next let ABC, DEF (fig. 9) represent three points in the one object 
and three points in the other object, the figure being drawn to a scale so that 
all the lines in the figure are the actual lines multiplied by /Xj. The lines of 
the figure represent the reduced paths of the rays between the corresponding 
points of the objects. 

Now it may be shown that the form of this figure cannot be altered with- 
out altering the length of one or more of the nine lines joining the points ABC 
to DEF. Therefore since the reduced paths of the rays in the image are equal 
to those in the object, the figure must represent the image on a scale of /n, 
to 1, and therefore the instrument must magnify every part of the object alike 
and elongate the distances parallel to the axis in the same proportion. It is 
therefore a telescope, and m=l. 

If iJi, = ix,, the image is exactly equal to the object, which is the case in 
reflexion in a plane mirror, which we know to be a perfect instrument for all 
distances. 

The only case in which by refraction at a single surface we can get a 
perfect image of more than one point of the object, is when the refracting 
surface is a sphere, radius r, index /x, and when the two objects are spherical 

surfaces, concentric with the sphere, their radii being - , and r ; and the two 

images also concentric spheres, radii /ar, and r. 

In this latter case the image is perfect, only at these particular distances 
and not generally. 

I am not aware of any other case in which a perfect image of an object 
can be formed, the rays being straight before they enter, and after they emerge 
from the instrument. The only case in which perfect astigmatism for all pencils 
has hitherto been proved to exist, was suggested to me by the consideration 



VOL. I. PLATE IV. 



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ON THE GENERAL LAWS OF OPTICAL INSTRUMENTS. 285 

of the structure of the crystalline lens in fish, and was published in one of 
the problem-papers of the Camhiidge and Dublin Mathematical Journal. My 
own method of treating that problem is to be found in that Journal, for 
February, 1854. The case is that of a medium whose index of refraction varies 
with the distance from a centre, so that if fi, be its value at the centre, a 
a given line, and r the distance of any point where the index is /x, then 



/^ = /Ao 



a' + r'' 



The path of every ray within this medium is a circle in a plane passing through 
the centre of the medium. 

Every ray from a point in the medium, distant b from the centre, will 

converge to a point on the opposite side of the centre and distant from it ^ . 

It will be observed that both the object and the image are included in 
the variable medium, otherwise the images would not be perfect. This case 
therefore forms no exception to the result of Prop. IX., in which the object and 
image are supposed to be outside tho instrument. 

Aberdeen, 12th Jan., 1858. 



[From the Proceedings of the Royal Society of Edinburgh, Vol. rv.] 



XYIII. On Theories of the Constitution of Saturn's Rings. 

The planet Saturn is surrounded by several concentric flattened rings, which 
appear to be quite free from any connection with each other, or with the planet, 
except that due to gravitation. 

The exterior diameter of the whole system of rings is estimated at about 
176,000 miles, the breadth from outer to inner edge of the entire system, 
36,000 miles, and the thickness not more than 100 miles. 

It is evident that a system of this kind, so broad and so thin, must 
depend for its stability upon the dynamical equihbrium between the motions of 
each part of the system, and the attractions which act on it, and that the 
cohesion of the parts of so large a body can have no effect whatever on its 
motions, though it were made of the most rigid material known on earth. It 
is therefore necessary, in order to satisfy the demands of physical astronomy, 
to explain how a material system, presenting the appearance of Saturn's Kings, 
can be maintained in permanent motion consistently with the laws of gravitation. 
The principal hypotheses which present themselves are these — 
I. The rings are solid bodies, regular or irregular. 
II. The rings are fluid bodies, liquid or gaseous. 
in. The rings are composed of loose materials. 

The results of mathematical investigation appHed to the first case are, — 

1st. That a uniform ring cannot have a permanent motion. 

2nd. That it is possible, by loading one side of the ring, to produce 
stability of motion, but that this loading must be very great compared with 
the whole mass of the rest of the ring, being as 82 to 18. 



ON THEORIES OF THE CONSTITUTION OF SATURN's RINGS, 287 

3rd. That this loading must not only be very great, but very nicely 
adjusted; because, if it were less than '81, or more than 83 of the whole, 
the motion would be unstable. 

The mode in which such a system would be destroyed would be by the 
collision between the planet and the inside of the ring. 

And it is evident that as no loading so enormous in comparison with the 
ring actually exists, we are forced to consider the rings as fluid, or at least 
not solid ; and we find that, in the case of a fluid ring, waves would be gene- 
rated, which would break it up into portions, the number of which would 
depend on the mass of Saturn directly, and on that of the ring inversely. 

It appears, therefore, that the only constitution possible for such a ring is 
a series of disconnected masses, which may be fluid or solid, and need not be 
equal. The \iomplicated internal motions of such a ring have been investigated, 
and found to consist of four series of waves, which, when combined together, 
will reproduce any form of original disturbance with all its consequences. The 
motion of one of these waves was exhibited to the Society by means of a small 
mechanical model made by Ramage of Aberdeen. 

This theory of the rings, being indicated by the mechanical theory as the 
only one consistent with permanent motion, is further confirmed by recent obser- 
vations on the inner obscure ring of Saturn. The limb of the planet is seen 
through the substance of this ring, not refracted, as it would be through a 
gas or fluid, but in its true position, as would be the case if the light passed 
through interstices between the separate particles composing the ring. 

As the whole investigations are shortly to be published in a separate form, 
the mathematical methods employed were not laid before the Society. 



XIX. On the Stability of the motion of Saturn's Rings. 



[An Essay, which obtained the Adams Prize for the year 1856, in the University 

of Cambridge.] 



ADVERTISEMENT. 

The Subject of the Prize was announced in the following terms ; — 

The University having accepted a fimd, raised by several members of St John's Collegp, 
for the purpose of foun ding a Prize to be called the Adams Prize, for the best Essay 
on some subject of Pure Mathematics, Astronomy, or other branch of Natural Pliilosophy, 
the Prize to be given once in two years, and to be open to tlhe competition of all persons 
who have at any time been admitted to a degree in this University: — 

The Examiners give Notice, that the following is the subject for the Prize to be adjudged 
in 1857:— 

The Motions of iSaturn's Rings. 

*** The problem may be treated on the supposition that the system of Rings is exactly or 
very approximately concentric with Saturn and symmetrically disposed about the plane of his Equator, 
and different hypotheses may be made respecting the physical constitution of the Rings. It may 
be supposed (1) that they are rigid: (2) that they ai-e fluid, or in part aeriform: (3) that they 
consist of masses of matter not mutually coherent. The question will be considered to be answered 
by ascertaining on tliese hypotheses severally, whether the conditions of mechanical stability are 
satisfied by the mutual attractions and motions of the Planet and the Rings. 

It is desirable that an attempt should also be made to determine on which of the above 
hypotheses the appearances both of the bright Rings and the recently discovered dark Ring may 
be most satisfactorily explained; and to indicate any causes to which a change of form, such as 
is supposed from a comparison of modern with the earlier observations to have taken place, may 
be attributed. 

E. GUEST, rice-Chancellor. 

J. CHALLIS. 

S. PARKINSON. 

W. THOMSON. 
March 23, 1855. 



ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 289 



CONTENTS. 

Nature of the Problem 290 

Laplace's investigaticm of the Equilibrium of a Ring, otuI its minimum density 29i 

Hit proof that Oie platie of tlie Rings will follow that of Saturn's Equator — that a solid uniform Ring is 

necessarily unstable 293 

Further investigation required— Theory of an Irregular Solid Ring leads to the result that to ensure stability 

the irregularity mitst be enormous 294 

Theory of a fluid or discontinuous Ring resolves itself into the investigation of a series of waves . . 295 

PART I. 

ON THE MOTION OF A RIGID BODY OF ANY FORM ABOUT 
A SPHERE. 

Equations of Motion 296 

Problem I. To find the conditions under which a uniform motion of the Ring is possible . . . 298 

Problem II. To find the equations of the motion when slightly distxirbed 299 

Problem III. To reduce the three siinultaneou^ equations of motion to the form of a single linear equation 300 
Problem IV. To determijie whether the motion of the Ring is stable or unstable, by meayis of the relations 

of the coefficients A, B, C 301 

Problem V. To find the centre of gravity, the radius of gyration, and the variations of the potential Tieaf 

the centre of a circular ring of small but variable section 302 

Problem VI. To determine the condition of stability of the motion in terms of the coeffilcierits f, g, h, which 

indicate the distribution of mass in the ring 306 

RB6ULTS. I«^, a uniform ring is unstable. 2nd, a ring varying in section according to the law of sines is 
unstable. 3rd, a uniform ring loaded with a heavy particle may be stable, provided the mass of the 
particle be between 'SlSSeS and "8279 of the whole. Case in which the ring is to the particle as 18 
«o 82 307 

PART II. 

ON THE MOTION OF A RING, THE PARTS OF WHICH ARE NOT 
RIGIDLY CONNECTED. 

1. General Statemeiii of the Problem, and limitation to a nearly uniform ring 310 

2. Notation 311 

3. Expansion of a function in terms of sines and cosines of multiples of the variable . . . . 311 

4. Magnitude and direction of attraction between two elements of a disturbed Ring 312 

5. Resultant attractions on any one of a ring of equal satellites disturbed in any way .... 313 
Note. Calculated values of these attractions in particular cases . . . . . . . . 314 

6. Equations of motion of a satellite of the Ring, and biquadratic equation to determine the wave-velocity 31.5 

7. A ring of satellites may always be rendered stable by increasing the mass of the central body . . 317 

8. Relation between the number and mass of satellites and the mass of the central body necessary to 

ensure stability. S>-4352,x2R 318 

9. Solution of the biquadratic equation when the mass of t/ie Ring is small ; and complete e.rprcssions 

for the Tnotion of each satellite .............. 319 

10. Each satellite moves {relatively to tJie ring) in an ellipse 321 



290 ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 

11. Each satellite moves absolutely/ in S2>ace in a curve which is nearly an ellipse for the large values 

of n, and a spiral of many nearly circular coih when n is small 321 

12. The form of the ring at a given instant is a series of undulations 322 

13. These uiidvlations travel round the ring with velocity relative to the ring, and a absolutely 323 

14. General Solution of the Problem — Given the position and motion of every satellite at any one time, 
to calculate the position and motion of any satellite at any other time, provided that the condition 

of stability is fulfilled 323 

15. Calculation of the effect of a periodic external disturbing force 326 

16. Treatment of disturbing forces in general 328 

17. Theory of free waves and forced waves 329 

18. Motion of the ring when the conditions of stability are net fulfilled. Two different ways in which 

the ring may be broken up 330 

19. Motion of a riyig of unequal satellites 335 

20. Motion of a ring composed of a clowi of scattered particles 336 

21. Calculation of the forces arising from the displacements of such a system 337 

22. Application to the case of a ring of this kind. The mean density must be excessively s^nall, which 

is iTiconsistent with its moving as a whole ............ 338 

23. On the forces arising from inequalities in a thin stratum of gravitating incompressible fluid of 
indefinite extent 338 

24. Application to the case of a flattened fluid ring, moving with uniform angular velocity. Such a 

ring will be broken up into portions which may continue to revolve as a ring of satellites . . . 344 

ON THE MUTUAL PERTUKBATIONS OF TWO RINGS. 

25. Application of the general theory of free and forced waves 345 

26. To determine the attractions between the rings 346 

27. To form the equations of motion 348 

28. Method of determining the reaction of the forced wave on tlve free wave which produced it . . 349 

29. Cases in which the perturbations increase indefinitely . . ........ 351 

30. Application to the theory of an indefinite number of concentric rings 352 

31. On the effect of long-continued disturbances on a system of rings 352 

32. On the effect of collisions among the parts of a revolving system ....... 354 

33. On the effect of internal friction in a fluid ring 354 

Ilecapitulation of the Theory of the Motion of a Rigid Ring. Reasons for ryecting tlie hypothesis of 

rigidity 356 

Recapitulation of the Theory of a Ring of Equ/il Satellites 360 

Description of a working model shewing the motions of such a system 363 

Theory of Rings of various constitutions 367 

Mutual action of Two Rings 370 

Case of many concentric Rings, &c. . . 371 

General Conclusions 372 

Appendix. Extract of a letter from Professor W. Thomson, of Glasgow, giving a solution of t/ie Pro- 
blem of a Rigid Ring 374 



There are some questions in Astronomy, to which we are attracted rather 
on account of their pecuHarity, as the possible illustration of some unknown 
principle, than from any direct advantage which their solution would afford to 



ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 291 

mankind. The theory of the Moon's inequalities, though in its first stages it 
presents theorems interesting to all students of mechanics, has been pursued into 
such intricacies of calculation as can be followed up only by those who make 
the improvement of the Lunar Tables the object of their lives. The value of 
the labours of these men is recognised by all who are aware of the importance 
of such tables in Practical Astronomy and Navigation. The methods by which 
the results are obtained are admitted to be sound, and we leave to professional 
astronomers the labour and the merit of developing them. 

The questions which are suggested by the appearance of Saturn's Rings 
cannot, in the present state of Astronomy, call forth so great an amount of 
labour among mathematicians. I am not aware that any practical use has been 
made of Saturn's Rings, either in Astronomy or in Navigation. They are too 
distant, and too insignificant in mass, to produce any appreciable effect on the 
motion of other parts of the Solar system; and for this very reason it is diflS- 
cult to determine those elements of their motion which we obtain so accurately 
in the case of bodies of greater mechanical importance. 

But when we contemplate the Rings from a purely scientific point of view, 
they become the most remarkable bodies in the heavens, except, perhaps, those 
still less useful bodies — the spiral nebulae. When we have actually seen that 
great arch swung over the equator of the planet without any visible connexion, 
we cannot bring our minds to rest. We cannot simply admit that such is the 
case, and describe it as one of the observed facts in nature, not admitting or 
requiring explanation. We must either explain its motion on the principles of 
mechanics, or admit that, in the Saturnian realms, there can be motion regu- 
lated by laws which we are unable to explain. 

The arrangement of the rings is represented in the figure (l) on a scale 
of one inch to a hundred thousand miles. S is a section of Saturn through 
his equator, A, B and C are the three rings. A and B have been known for 
200 years. They were mistaken by Galileo for protuberances on the planet itself, 
or perhaps satellites. Huyghens discovered that what he saw was a thin flat 
ring not touching the planet, and Ball discovered the division between A and B. 
Other divisions have been observed splitting these again into concentric rings, 
but these have not continued visible, the only well-established division being one 
in the middle of A. The third ring C was first detected by Mr Bond, at 
Cambridge U.S. on November 15, 1850; Mr Dawes, not aware of Mr Bond's 
discovery, observed it on November 29th, and Mr Lassel a few days later. It 



292 ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 

gives little light compared with the other rings, and is seen where it crosses 
the planet as an obscure belt, but it is so transparent that the limb of the 
planet is visible through it, and this without distortion, shewing that the rays 
of light have not passed through a transparent substance, but between the 
scattered particles of a discontinuous stream. 

It is difficult to estimate the thickness of the system ; according to the 
best estimates it is not more than 100 miles, the diameter of A being 176,418 
miles; so that on the scale of our figure the thickness would be one thousandth 
of an inch. 

Such is the scale on which this magnificent system of concentric rings is 
constructed; we have next to account for their continued existence, and to 
reconcile it with the known laws of motion and gravitation, so that by rejecting 
every hypothesis which leads to conclusions at variance with the facts, we may 
learn more of the nature of these distant bodies than the telescope can yet 
ascertain. We must account for the rings remaining suspended above the planet, 
concentric with Saturn and in his equatoreal plane ; for the flattened figure of the 
section of each ring, for the transparency of the inner ring, and for the gradual 
approach of the inner edge of the ring to the body of Saturn as deduced 
from all the recorded observations by M. Otto Struvd {Sur les dimensions des 
Anneaux de Saturne — Recueil de Memoires Astronomiques, Poulkowa, 15 Nov. 
1851). For an account of the general appearance of the rings as seen from the 
planet, see Lardner on the Uranography of Saturn, Mem. of the Astronomical 
Society, 1853. See also the article "Saturn" in Nichol's Cyclopcedia of the 
Physical Sciences. 

Our curiosity with respect to these questions is rather stimulated than 
appeased by the investigations of Laplace. That great mathematician, though 
occupied with many questions which more imperiously demanded his attention, 
has devoted several chapters in various parts of his great work, to points con- 
nected with the Saturnian System. 

He has investigated the law of attraction of a ring of small section on a 
point very near it {Mec. Cel. Liv. iii. Chap, vi.), and from this he deduces the 
equation from which the ratio of the breadth to the thickness of each ring is 

to be found, 

E' p X(X-l) 

^~3a'p (\+l) (3X^+1)' 

where R is the radius of Saturn, and p his density; a the radius of the ring, 



ON THE STABIUTY OF THE MOTION OF SATURN's RINGS. 293 

and p its density; and X the ratio of the breadth of the ring to its thick- 
ness. The equation for determining X when e is given has one negative root 
which must be rejected, and two roots which are positive while e<0"0543, and 
impossible when e has a greater value. At the critical value of e, X = 2-594 
nearly. 

The fact that X is impossible when e is above this value, shews that the 
ring cannot hold together if the ratio of the density of the planet to that of 
the ring exceeds a certain value. This value is estimated by Laplace at I'S, 
assuming a = 2R. 

We may easily follow the physical interpretation of this result, if we observe 
that the forces which act on the ring may be reduced to — 

(1) The attraction of Saturn, varying inversely as the square of the dis- 
tance from his centre. 

(2) The centrifugal force of the particles of the ring, acting outwards, and 
varying directly as the distance from Saturn's polar axis. 

(3) The attraction of the ring itself, depending on its form and density, 
and directed, roughly speaking, towards the centre of its section. 

The first of these forces must balance the second somewhere near the mea,n 
distance of the ring. Beyond this distance their resultant will be outwards, 
within this distance it will act inwards. 

If the attraction of the ring itself is not sufl&cient to balance these residual 
forces, the outer and inner portions of the ring will tend to separate, and the 
ring will be split up ; and it appears from Laplace's result that this will be 
the case if the density of the ring is less than ^ of that of the planet. 

This condition applies to all rings whether broad or narrow, of which the 
parts are separable, and of which the outer and inner parts revolve with the 
same angular velocity. 

Laplace has also shewn (Li v. v. Chap, iii.), that on account of the oblate- 
ness of the figure of Saturn, the planes of the rings will follow that of Saturn's 
equator through every change of its position due to the disturbing action of 
other heavenly bodies. 

Besides this, he proves most distinctly (Liv. iii. Chap, vi.), that a solid uni- 
form ring cannot possibly revolve about a central body in a permanent manner, 
for the slightest displacement of the centre of the ring from the centre of the 
planet would originate a motion which would never be checked, and would 



294 ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 

inevitably precipitate the ring upon the planet, not necessarily by breaking the 
ring, but by the inside of the ring falling on the equator of the planet. 

He therefore infers that the rings are irregular solids, whose centres of 
gravity do not coincide with their centres of figure. We may draw the con- 
clusion more formally as follows, "If the rings were solid and uniform, their 
motion would be unstable, and they would be destroyed. But they are not 
destroyed, and their motion is stable; therefore they are either not uniform or 
not solid." 

I have not discovered"" either in the works of Laplace or in those of more 
recent mathematicians, any investigation of the motion of a ring either not uni- 
form or not solid. So that in the present state of mechanical science, we do 
not know whether an irregular solid ring, or a fluid or disconnected ring, can 
revolve permanently about a central body; and the Saturnian system still re- 
mains an unregarded witness in heaven to some necessary, but as yet unknown, 
development of the laws of the universe. 

We know, since it has been demonstrated by Laplace, that a uniform solid 
ring cannot revolve permanently about a planet. We propose in this Essay to 
determine the amount and nature of the irregularity which would be required 
to make a permanent rotation possible. We shall find that the stability of the 
motion of the ring would be ensured by loading the ring at one point with a 

* Since this -was written, Prof. Challis has pointed out to me three important papers in Gould's 
Astronomical Journal: — Mr G. P. Bond on the Rings of Saturn (May 1851) and Prof. B. Pierce of 
Harvard University on the Constitution of Saturn's Rings (June 1851), and on the Adams' Prize 
Problem for 1856 (Sept. 1855). These American mathematicians have both considered the conditions 
of statical equilibrium of a transverse section of a ring, and have come to the conclusion that the 
rings, if they move each as a whole, must be very narrow compared with the observed rings, so 
that in reality there must be a great number of them, each revolving with its own velocity. They 
have also entered on the question of the fluidity of the rings, and Prof. Pierce has made an 
investigation as to the permanence of the motion of an irregular solid ring and of a fluid ring. 
The paper in which these questions are treated at large has not (so far as I am aware) been 
pxiblished, and the references to it in Gould's Journal are intended to give rather a popular account 
of the results, than an accurate outline of the methods employed. In treating of the attractions of 
an irregular ring, he makes admirable use of the theory of potentials, but his published investi- 
gation of the motion of such a body contains some oversights which are due perhaps rather to the 
imperfections of popular language than to any thing in the mathematical theory. The only part of 
the theory of a fluid ring which he has yet given an account of, is that in which he considers 
the form of the ring at any instant as an ellipse; corresponding to the case where n = u), and 
m=l. As I had only a limited time for reading these papers, and as I could not ascertain the 
methods used in the original investigations, I am unable at present to state how far the results of 
this essay agree with or differ from those obtained by Prof. Pierce. 



ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 295 

heavy satellite about 4-^ times the weight of the ring, but this load, besides 
being inconsistent with the observed appearance of the rings, must be far too 
artificially adjusted to agree with the natural arrangements observed elsewhere, 
for a very small error in excess or defect would render the ring again unstable. 

We are therefore constrained to abandon the theory of a solid ring, and 
to consider the case of a ring, the parts of which are not rigidly connected, 
as in the case of a ring of independent satellites, or a fluid ring. 

There is now no danger of the whole ring or any part of it being pre- 
cipitated on the body of the planet. Every particle of the ring is now to be 
regarded as a satellite of Saturn, disturbed by the attraction of a ring of 
satellites at the same mean distance from the planet, each of which however is 
subject to slight displacements. The mutual action of the parts of the ring will 
be so small compared with the attraction of the planet, that no part of the 
ring can ever cease to move round Saturn as a satellite. 

But the question now before us is altogether different from that relating to 
the solid ring. We have now to take account of variations in the form and 
arrangement of the parts of the ring, as well as its motion as a whole, and 
we have as yet no security that these variations may not accumulate till the 
ring entirely loses its original form, and collapses into one or more satellites, 
circulating round Saturn. In fact such a result is one of the leading doctrines 
of the " nebular theory " of the formation of planetary systems : and we are 
familiar with the actual breaking up of fluid rings under the action of "capil- 
lary " force, in the beautiful experiments of M. Plateau. 

In this essay I have shewn that such a destructive tendency actually exists, 
but that by the revolution of the ring it is converted into the condition of 
dynamical stability. As the scientific interest of Saturn's Rings depends at 
present mainly on this question of their stability, I have considered their motion 
rather as an illustration of general principles, than as a subject for elaborate 
calculation, and therefore I have confined myself to those parts of the subject 
which bear upon the question of the permanence of a given form of motion. 

There is a very general and very important problem in Dynamics, the solu- 
tion of which would contain all the results of this Essay and a great deal 
more. It is this — 

"Having found a particular solution of the equations of motion of any 
material system, to determine whether a slight disturbance of the motion indi- 



296 ON THE STABILITY OF THE MOTION OF SATURN S RINGS. 

cated by the solution would cause a small periodic variation, or a total 
derangement of the motion." 

The question may be made to depend upon the conditions of a maximum 
or a minimum of a function of many variables, but the theory of the tests 
for distinguishing maxima from minima by the Calculus of Variations becomes 
so intricate when applied to functions of several variables, that I think it doubt- 
ful whether the physical or the abstract problem will be first solved. 



PART I. 

ON THE MOTION OF A RIGID BODY OF ANY FORM ABOUT A SPHERE. 

We confine our attention for the present to the motion in the plane of 
reference, as the interest of our problem belongs to the character of this motion, 
and not to the librations, if any, from this plane. 

Let S (Fig. 2) be the centre of gravity of the sphere, which we may call 
Satiu-n, and E that of the rigid body, which we may call the Ring. Join RS, 
and divide it in G so that 

SG : GR '.: R : S, 

R and S being the masses of the Ring and Saturn respectively. 

Then G will be the centre of gravity of the system, and its position will 
be unaffected by any mutual action between the parts of the system. Assume G 
as the point to which the motions of the system are to be referred. Draw GA 
in a direction fixed in space. 

Let AGR = e, and SR = r, 

then ^^^'S+R^' ^^^ ^^^STR^' 

so that the positions of S and R are now determined. 

Let BRR be a straight line through R, fixed with respect to the substance 
of the ring, and let BRK=^. 



ON THE STABILITY OF THE MOTION OF SATURN S RINGS. 297 

This determines the angular position of the ring, so that from the values 
of r, 6, and ^ the configuration of the system may be deduced, as far as relates 
to the plane of reference. 

We have next to determine the forces which act between the ring and 
the sphere, and this we shall do by means of the potential function due to 
the ring, which we shall call V. 

The value of V for any point of space S, depends on its position relatively 
to the ring, and it is found from the equation 



where dm is an element of the mass of the ring, and r is the distance of that 
element from the given point, and the summation is extended over every element 
of mass belonging to the ring. V will then depend entirely upon the position 
of the point S relatively to the ring, and may be expressed as a function 
of r, the distance of S from R, the centre of gravity of the ring, and ^, the 
angle which the line SR makes with the line RB, fixed in the ring. 

A particle P, placed at S, will, by the theory of potentials, experience a 

dV . ... . \ dV 

moving force P —p in the direction which tends to increase r, and P - -jj 

in a tangential direction, tending to increase ^. 

Now we know that the attraction of a sphere is the same as that of 

a particle of equal mass placed at its centre. The forces acting between the 

dV . . 

sphere and the ring are therefore S -j~ tending to increase r, and a tangential 

\ dV . 

force S - -j-r , applied at S tending to increase <;^. In estimating the efiect of 

this latter force on the ring, we must resolve it into a tangential force S - -jj- 

dV 
acting at R, and a couple S -j-r tending to increase (f). 

We are now able to form the equations of motion for the planet and the 
ring. 



298 ON THE STABILITY OF THE MOTION OF SATURN S RINGS. 

For the planet 



^d jf Rr Ydd\ _R_ ^Jy , . 

^ dt ]S^VRl dtj '- " S + R '^ d<f> ^'^' 

«l(^)-^(f)'=^^' (^)- 

For the centre of gravity of the ring, 

j.d (f Sr Y ^^1 S dV , . 

^dt\\S+-R) Ttr~STR^df ^ ^' 

j.d^ f Sr \ Sr (d0Y_ dV , . 

For the rotation of the ring about its centre of gravity, 

^S(''+«=^f (5)' 

where h is the radius of gyration of the ring about its centre of gravity. 

Equation (3) and (4) are necessarily identical with (l) and (2), and shew 
that the orbit of the centre of gravity of the ring must be similar to that 
of the Planet. Equations (1) and (3) are equations of areas, (2) and (4) are 
those of the radius vector. 

Equations (3), (4) and (5) may be thus written, 

M-'^T!-'-'^}-(^-^i'- («)' 

-{§-©}-(---)f - (^)- 

-(f-^f)--^ - («)• 

These are the necessary and sufficient data for determining the motion of 
the ring, the initial circumstances being given. 

Prob. I. To find the conditions under which a uniform motion of the 
ring is possible. 

By a uniform motion is here meant a motion of uniform rotation, during 
which the position of the centre of the Planet with respect to the ring does 
not change. 



ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 299 

In this case r and </> are constant, and therefore V and its differential 
coefficients are given. Equation (7) becomes, 

which shews that the angular velocity is constant, and that 

dey R+S dV , ,^. 

r- = <o\ say (9). 



dtj Rr dr 

(PB 
Hence, -71 = 0, and therefore by equation (8), 

%-^ • •■•••(-)• 

Equations (9) and (10) are the conditions under which the uniform motion 
is possible, and if they were exactly fulfilled, the uniform motion would go on 
for ever if not disturbed. But it does not follow that if these conditions were 
nearly fulfilled, or that if when accurately adjusted, the motion were slightly 
disturbed, the motion would go on for ever nearly uniform. The effect of the 
disturbance might be either to produce a periodic variation in the elements 
of the motion, the ampUtude of the variation being small, or to produce a 
displacement which would increase indefinitely, and derange the system altogether. 
In the one case the motion would be dynamically stable, and in the other it 
would be dynamically unstable. The investigation of these displacements while 
still very small wiU form the next subject of inquiry. 

Prob. II. To find the equations of the motion when slightly disturbed. 
Let r = r„ = o}t and (f) = (f>^ in. the case of uniform motion, and let 

r=ro +r„ 

e=a)t+e„ 

when the motion is slightly disturbed, where r^, 6^, and ^1 are to be treated 

as small quantities of the first order, and their powers and products are to be 

dV dV 

neglected. We may expand -j-^ and -j-r by Taylor's Theorem, 

dV_dV drV d'V 

dr ~dr "^ di^ '''"*■ cZrc/t^"^^' 

d<f>~'d<f'^drd<t>''''^ d<i>''^'' 



300 ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 

where the values of the differential coeflBcients on the right-hand side of the 
equations are those in which i\ stands for r, and ^^ for ^. 

CaJlmg ^=A ^^^ = M^ ^^=N, 

and taking account of equations (9) and (10), we may write these equations, 

a^= -sirs'" +^''+^^" 

Substituting these values in equations (6), (7), (8), and retaining all small 
quantities of the first order while omitting their powers and products, we have 
the following system of linear equations in r^, O^, and ^i, 

E (2r,co^ + r,^^^y{E + S)(Mr, + N<f.,) =0 (11), 



R 



d% , „ de\ 



df 



(o%-2r,(o-^]-{R + S){L7\ + M<f>,) = (12), 



RlH'^^ + ^-SiMr^ + N^:) =0 (13). 



df ' df 



Prob. III. To reduce the three simultaneous equations of motion to the 
form of a single linear equati 



:ion. 



Let us write n instead of the symbol -j- , then arranging the equations in 
terms of i\, 6^, and j>^, they may be written: 

{2R,o>n + (R + S)M}r, + (Rr:n')e, + {R + S)N<i>, =0 (14), 

{Rn'-R<^'^-(R + S) L}r,-(2Rr,con)d,^{R + S)M<f>, = (15), 

- (SM) r, + (Rk'n') 0, + {RUrv -SN)<j>, =0 (16). 

Here we have three equations to determine three quantities r,, 6„ ^i ; but 
it is evident that only a relation can be determined between them, and that 
in the process for finding their absolute values, the three quantities will vanish 
together, and leave the following relation among the coefiicients, 



ON THE STABILITY OF THE MOTION OF SATURN'S RINGS. 301 

-{2Rr,oin+ {R + S)^r} [2R)\(on] [Rlcrc'-SN} 
+ {Rn' - Rco' -(R + S) L] {Rh'rf} {(R + >S') N] 

+ {SM) {Rrjn') {R + S)M- (SM) {2Rr,<on) (R + S)Xi=0 (17). 

+ {2Rr,<on + (R + S)M} {RLni'} {(R + S) if} 
- {Rn' - Rxo' -{R + S)} {Rr.'if} {RJc'n' - SN} 
By multiplying up, and arranging by powers of n and dividing by Rn\ 
this equation becomes 

Aii* + B)v+C=0 (18), 

where 

B = SRr-r:i''<o-'-R{R + S)Lr:Jc'-R{{R + S)]if + Si''}N- i (19). 

C=R{(R + S)l''- 3Sr:} oy + (R + S) {{R + S) t + Sr^} (Z.V- IP) J 
Here we have a biquadratic equation in ?i which may be treated as a 
quadratic in ?r, it being remembered that ?i stands for the operation -j- . 

Prob. IV. To determine whether the motion of the ring is stable or 
unstable, by means of the relations of the coefficients A, B, C. 

The equations to determine the forms of r^, 6^, and <^i are all of the form 
. d*u -r, dhi ^ ^ /^^\ 

^*+-^*+^"=» (-°'' 

and if n be one of the four roots of equation (18), then 

will be one of the four terms of the solution, and the values of i\, 6^, and 
<^i will differ only in the values of the coefficient D. 

Let us inquire into the nature of the solution in different cases. 

(1) If n be positive, this term would indicate a displacement which 
must increase indefinitely, so as to destroy the arrangement of the system. 

(2) If n be negative, the disturbance which it belongs to would gradually 
die away. 

(3) If n be a pure impossible quantity, of the form ±aj —\, then there 
will be a term in the solution of the form D cos [at + a), and this would indi- 

277 

cate a periodic variation, whose amplitude is D, and period ^^ . 



302 ON THE STABILITY OF THE MOTION OF SATURN S RINGS. 

(4) If n be of the form b±J'^a, the first term being positive and 
the second impossible, there will be a term in the solution of the form 

De^' cos {at + a), 
which indicates a periodic disturbance, whose amplitude continually increases 
till it disarranges the system. 

(5) If n be of the form -h±s/-la, a negative quantity and an im- 
possible one, the corresponding term of the solution is 

i>e"*'cos {(it + a), 
which indicates a periodic disturbance whose amplitude is constantly diminishing. 

It is manifest that the first and fourth cases are inconsistent with the 
permanent motion of the system. Now since equation (18) contains only even 
powers of n, it must have pairs of equal and opposite roots, so that every 
root coming under the second or fifth cases, implies the existence of another 
root belonging to the first or fourth. If such a root exists, some disturbance 
may occur to produce the kind of derangement corresponding to it, so that 
the system is not safe unless roots of the first and fourth kinds are altogether 
excluded. This cannot be done without excluding those of the second and fifth 
kinds, so that, to insure stability, aU the four roots must be of the third kind, 
that is, pure impossible quantities. 

That this may be the case, both values of n" must be real and negative, 
and the conditions of this are — 

1st. That A, B, and C should be of the same sign, 

2ndly. That R>iAC. 

When these conditions are fulfilled, the disturbances will be periodic and 
consistent with stability. When they are not both fulfilled, a small disturbance 
may produce total derangement of the system. 

Prob. V. To find the centre of gravity, the radius of gyration, and the 
variations of the potential near the centre of a circular ring of small but variable 
section. 

Let a be the radius of the ring, and let 6 be the angle subtended at the 
centre between the radius through the centre of gravity and the line through 
a given point in the ring. Then if /i be the mass of unit of length of the 



ON THE STABILITY OF THE MOTION OF SATURn's RINGS. 303 

ring near the given point, ft will be a periodic function of 6, and may there- 
fore be expanded by Fourier's theorem in the series, 

li = — {1 + 2/cos^ + §^cos2^ + §/isin2^ + 2ico3(3^ + a) + &c.} (21), 

where/, g, h, &c. are arbitrary coefficients, and R is the mass of the ring. 

(1) The moment of the ring about the diameter perpendicular to the 
prime radius is 

R)\= r ficr cos ecW = Raf, 
therefore the distance of the centre of gravity from the centre of the ring, 

(2) The radius of gyration of the ring about its centre in its own plane 
is evidently the radius of the ring =a, but if k be that about the centre of 
gravity, we have 

.'. Af = a=(l-f). 

(3) The potential at any point is found by dividing the mass of each 
element by its distance from the given point, and integrating over the whole 
mass. 

Let the given point be near the centre of the ring, and let its position be 
defined by the co-ordinates r and xjj, of which r is small compared with a. 

The distance (p) between this point and a point in the ring is 
i = i {1 + %03 (^ - 0) + i (Q' + 1 (3' cos 2{i,-0)+&c.}. 



The other terms contain powers of — higher than the second. 
We have now to determine the value of the integral, 

Jo P 
and in multiplying the terms of (/i) by those of f-J , we need retain only 
those which contain constant quantities, for all those which contain sines or 



304 ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 

cosines of multiples of {^1^ — 0) will vanisti when integrated between the limits. 
In this way we find 

^=- {l+/%osr/; + i^'(l-4-5rcos2i/, + ^sin2tA)} (22). 

The other terms containing higher powers of — . 

In order to express V in terms of r, and (f)„ as we have assumed in the 
former investigation, we must put 

r' C09 xjj= — Tj + ^r^^/, 

^=§{^-f'i^it^^+9) + i^fr.<f>. + ir<l>n^-9)} (23). 



From which we find , , 
dr 



'^.-s^- 



S.='^=i'(i+^) 






K).=^=i^'(^-^) 



These results may be confirmed by the following considerations applicable to 
any circular ring, and not involving any expansion or integration. Let af be 
the distance of the centre of gravity from the centre of the ring, and let 
the ring revolve about its centre with velocity o). Then the force necessary 
to keep the ring in that orbit will be —Rafoi^. 

But let >S be a mass fixed at the centre of the ring, then if 

a' 
every portion of the ring will be separately retained in its orbit by the attrac- 
tion of S, so that the whole ring will be retained in its orbit. The resultant 
attraction must therefore pass through the centre of gravity, and be 

-^ a} 

therefore ^^^rL, 

dr a: 



ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 305 

cl'V cfV d'V 
The equation 3^ + rf^- + dz' + *'P = « 

is true for any system of matter attracting according to the law of gravitation. 
If we bear in mind that the expression is identical in form with that which 
measures the total efflux of fluid from a differential element of volume, where 

-J- , -J- , -7- are the rates at which the fluid passes through its sides, we may 

easily form the equation for any other case. Now let the position of a point 
in space be determined by the co-ordinates r, ^ and z, where z is measured 
perpendicularly to the plane of the angle <j>. Then by choosing the directions 
of the axes x, y, z, so as to coincide with those of the radius vector r, the per- 
pendicular to it in the plane of <^, and the normal, we shall have 
dx = dr^ dy = rd^, dz = dz, 

dV^dV dV^ldV dV^dV 
dx~ dr ^ dy r d<l>' dz dz 

The quantities of fluid passing through an element of area in each direction are 
-T- rd(paz, -j-7 - ardz, -p rdcpdr, 

so that the expression for the whole efflux is 

1 dV d^V 1 d^V d^V 

r dF^d^^7 df^d^ ^^^' 

which is necessarily equivalent to the former expression. 

d^V 
Now at the centre of the ring -r^ may be found by considering the attrac- 
tion on a point just above the centre at a distance z, 

dV_ p z 

dz {a'->tz'f' 

d'V R . 

-^=--3,whenz = 0. 

Ai 1 \ dV R , . 

Also we know ^ = — ^ , and r = aj, 

V (XV (Xi 

so that m any curcular rmg "^^^^ d^^ a^ ^ ** 

an equation satisfied by the former values of L and N. 



306 ON THE STABILITY OF THE MOTION OF SATURN S RINGS. 

By referring to tlae original expression for the variable section of the ring, 
it appears that the effect of the coefficient / is to make the ring thicker on 
one side and thinner on the other in a uniformly graduated manner. The eflfect 
of ^ is to thicken the ring at two opposite sides, and diminish its section in 
the parts between. The coefficient h indicates an inequality of the same kind, 
only not symmetrically disposed about the diameter through the centre of 
gravity. 

Other terms indicating inequalities recurring three or more times in the 
circumference of the ring, have no effect on the values of X, M and N. There is 
one remarkable case, however, in which the irregularity consists of a single 
heavy particle placed at a point on the circumference of the ring. 

Let P be the mass of the particle, and Q that of the uniform ring on 
which it is fixed, then R = P-{-Q, 

■> K' 



-^S-^.=^(-^S=.4(-^) 



••• 3 = ^ = 3/- (27)- 

Prob. VI. To determine the conditions of stability of the motion in terms 
of the coefficients/, g, h, which indicate the distribution of mass in the ring. 

The quantities which enter into the differential equation of motion (18) 
are R, S, k", i\, (o", L, M, N. We must observe that S is very large compared 
with R, and therefore we neglect R in those terms in which it is added to S, 
and we put 



ON THE STABILITY OF THE MOTION OF SATURN S RINGS. 307 

Substituting these values in equation (18) and dividing by H'a*/-, we obtain 
{l-P)n* + (l-y^ + y^g)nW + (^-&r-lg^-lh^ + 2fg)<.^ = (28). 

The condition of stability is that this equation shall give both values of n* 
negative, and this renders it necessary that all the coefficients should have the 
same sign, and that the square of the second should exceed four times the 
product of the first and third. 

(1) Now if we suppose the ring to be uniform, /, g and h disappear, 
and the equation becomes 

n' + nV + | = (29), 

which gives impossible values to n' and indicates the instability of a uniform 
ring. 

(2) If we make g and A = 0, we have the case of a ring thicker at one 
side than the other, and varying in section according to the simple law of sines. 
We must remember, however, that / must be less than ^, in order that the 
section of the ring at the thinnest part may be real. The equation becomes 

(l_/=),,* + (l.|/^)^V + (|-6/>* = (30). 

The condition that the third term should be positive gives 

/*<'375. 

The condition that n' should be real gives 

71/^-112/^ + 32 negative, 

which requires/" to be between "37445 and 1'2. 

The condition of stability is therefore that /^ should lie between 

•37445 and '375, 

but the construction of the ring on this principle requires that /- should be 
less than "25, so that it is impossible to reconcile this fonn of the ring with 
the conditions of stability. 

(3) Let us next take the case of a uniform ring, loaded with a heavy 
particle at a point of its circumference. We have then g = Sf, h = 0, and the 
equation becomes 

(l-/=)n^ + (l-|/^ + f/ViV+(|-y/'+6/>^ = (31). 



308 ON THE STABILITY OF THE MOTION OF SATURN 3 RINGS. 

Dividing each term by 1 -/, we get 

(l+/)n^+(l+/-f/0^^V + f{3(l+/)-8/=}a,^ = O (32). 

The first condition gives /less than '8279. 

The second condition gives / greater than '8 15865. 

Let us assume as a particular case between these limits /= •82, which 
makes the ratio of the mass of the particle to that of the ring as 82 to 18, 
then the equation becomes 

l-82 7i^ + '8114?iV+-9696a>' = (33), 

which gives >J^^n= ±'5916(o or ±-3076w. 

These values of n indicate variations of r^, O^, and ^i, which are com- 
pounded of two simple periodic inequalities, the period of the one being 1"69 
revolutions, and that of the other 3 '2 51 revolutions of the ring. The relations 
between the phases and ampUtudes of these inequalities must be deduced from 
equations (14), (15), (16), in order that the character of the motion may be 
completely determined. 

Equations (14), (15), (16) may be written as follows: 

{Anco + hoi') ^ +2f7i%+f(3-g) (o"'(l>, = (34), 

{ii^-l<o'^{S+g)}^'-2fcone,^ifh<o'<f>, = (35), 

-/ho>^ '^ + 2 (1 -f^)n% + {2 (1 -f) n'-r {S-g) co^}<l>, = (36). 

By eliminating one of the variables between any two of these equations, 
we may determine the relation between the two remaining variables. Assuming 
one of these to be a periodic function of t of the form A cos pt, and remem- 
bering that n stands for the operation -7- , we may find the form of the other. 

Tlius, eliminating 6^ between the first and second equations, 

{n' + i7i<o'{5-g) + hoj'f-^+foy'{{3-g)<o-ym}cf>, = (37). 



ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 309 

T 

Assuming — =A^\wvt^ and <f)i = Q cos (ut — ^), 

{-v' + ^vo)' (5 - g)} A cos pt + h(o^ A sin vt +fo/ (3 -rj) Qcos{vt - /3) + Ifhui'vQ sin {yt - /3). 
Equating vt to 0, and to - , we get the equations 

[v'-^voy (5 -g)} A =f<o'Q {(3 -g) cj cos /8 - ^/ii/ sin /3}, 

- h<o' A =fo)'Q {(3 - </) o) sin /8 + -l/ii/ cos ^8}, 

from which to determine Q and ^. 

In all cases in which the mass is disposed symmetrically about the diameter 
through the centre of gravity, A = and the equations may be greatly simplified. 

Let 6i = P cos (vt — a), then the second equation becomes 

{v' + ^0)' (3 + g)} A sin vt = 2Pfa}v sin {vt - a), 

whence a = 0, P = ^^JtMiijO .4 (38). 

2j(DV ^ ' 

The first equation becomes 

^Aoiv cos vt - 2Pfv- cos vt + Qf (3 -g) w' cos (I'f - /S) = 0, 

whence ^ = 0, <? = '^"t.f ' w^^-^ (S^)- 

In the numerical example in which a heavy particle was fixed to the cir- 
cumference of the ring, we have, when /= '82, 



V 

^ 1-3076 



/•5916 P_r3-21 Q_f-l-229 
t-3076' A~\b-72' A~\- 797' 

so that if we put (ot = 0^ = the mean anomaly, 

^ = .4sin(-5916(9o-a)+^sin(-3076 6'o-^) (40), 



^1 = 3-21^ cos (-5916(90- a) + 5-72^ cos (-3070 ^0-/3) (41), 

<^,= -l-229^cos(-5916l9o-a)-5-7975cos(-30766',-/3) ... (42). 

These three equations serve to determine 1\, 6^ and <^i when the original 
motion is given. They contain four arbitrary constants A, B, a, /3. Now since 



310 ON THE STABILITY OF THE MOTION OF SATURN S RINGS. 

the original values 1\, 0^, <^i, and also their first differential coefficients with 
respect to t, are arbitrary, it would appear that six arbitrary constants ought 
to enter into the equation. The reason why they do not is that we assume 
r„ and 0^ as the Tiiean values of r and 6 in the actucd motion. These quantities 
therefore depend on the original circumstances, and the two additional arbitrary 
constants enter into the values of ^o and d^. In the analytical treatment of the 
problem the differential equation in n was originally of the sixth degree with a 
solution n- = 0, which implies the possibihty of terms in the solution of the 
form Ct + D. 

The existence of such terms depends on the previous equations, and we find 
that a term of this form may enter into the value of 6, and that r^ may contain 
a constant term, but that in both cases these additions will be absorbed into 
the values of 0, and r,. 



PART IL 

ON THE MOTION OF A RING, THE PARTS OF WHICH ARE NOT RIGIDLY CONNECTTED. 

1. In the case of the Ring of invariable form, we took advantage of the 
principle that the mutual actions of the parts of any system form at all times 
a system of forces in equilibrium, and we took no account of the attraction 
between one part of the ring and any other part, since no motion could result 
from this kind of action. But when we regard the different parts of the ring 
as capable of independent motion, we must take account of the attraction on 
each portion of the ring as affected by the irregularities of the other parts, and 
therefore we must begin by investigating the statical part of the problem in 
order to determine the forces that act on any portion of the ring, as depending 
on the instantaneous condition of the rest of the ring. 

In order to bring the problem within the reach of our mathematical methods, 
we limit it to the case in which the ring is nearly circular and uniform, and has 
a transverse section very small compared with the radius of the ring. By 
analysing the difficulties of the theory of a linear ring, we shall be better able 
to appreciate those which occur in the theory of the actual rings. 



ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 311 

The ring which we consider is therefore small in section, and very nearly 
circular and uniform, and revolving with nearly uniform velocity. The variations 
from circular form, uniform section, and uniform velocity must be expressed by a 
proper notation. 

2. To express the position of an element of a variable ring at a given time 
in terms of the original position of the element in the ring. 

Let S (fig. 3) be the central body, and SA a direction fixed in space. 

Let SB be a radius, revolving with the mean angular velocity w of the 
ring, so that ASB = (ot. 

Let n be an element of the ring in its actual position, and let P be the 
position it would have had if it had moved uniformly with the mean velocity w 
and had not been displaced, then BSP is a constant angle =s, and the value 
of 5 enables us to identify any element of the ring. 

The element may be removed from its mean position P in three different 
ways. 

(1) By change of distance from S by a quantity l^TT = p. 

(2) By change of angular position through a space Pp = a. 

(3) By displacement perpendicular to the plane of the paper by a quantity C 

p, a- and ^ are all functions of s and t. If we could calculate the attrac- 
tions on any element as depending on the form of these functions, we miglit 
determine the motion of the ring for any given original disturbance. We cannot, 
however, make any calculations of this kind without knowing the form of the 
functions, and therefore we must adopt the following method of separating the 
original disturbance into others of simpler form, first given in Fourier's Tmitc 
de Chaleur. 

3. Let C/" be a function of s, it is required to express U in a series of 
sines and cosines of multiples of s between the values 5 = and .s = 2t. 

Assume U=A,coss + A., cos 2*- + &c. -f A ^ cos nis + A „ cos ns 

+ B, sin ,s + B, cos 2.s + &c. + B,„ sin ms + B„ sin ns. 



312 ON THE STABILITY OF THE MOTION OP SATURN's RINGS. 

Multiply by coa Tusds and integrate, then all terms of the form 
J cos ms cos nsds and / cos ms sin nsds 
will vanish, if we integrate from s = to s = 27r, and there remains 

I U COS msds= IT A^, Ua\-D.msds = 'TrB^. 

If we can determine the values of these integrals in the given case, we 
can find the proper coefficients A^, B^, &c., and the series will then represent 
the values of U from s = to 5 = 27r, whether those values be continuous or 
discontinuous, and when none of those values are infinite the series will be 
convergent. 

In this way we may separate the most complex disturbances of a ring into 
parts whose form is that of a circular function of s or its multiples. Each of 
these partial disturbances may be investigated separately, and its efiect on the 
attractions of the ring ascertained either accurately or approximately. 

4. To find the magnitude and direction of the attraction between two 
elements of a disturbed ring. 

Let P and Q (fig. 4) be the two elements, and let their original positions 
be denoted by s^ and 5j, the values of the arcs BP, BQ before displacement. 
The displacement consists in the angle BSP being increased by ctj and BSQ 
by 0*2 , while the distance of P from the centre is increased by p, and that of 
Q by Pj. We have to determine the effect of these displacements on the distance 
PQ and the angle SPQ. 

Let the radius of the ring be unity, and 5j — .9i = 2^, then the original 
value of PQ will be 2 sin 0, and the increase due to displacement 

= (/>2 + Pi) sin ^ + (o-j - (Ti) cos 6. 

We may write the complete value of PQ thus, 

PQ = 2Bme{l+i{p, + p,)+^{(T,-(T,)cot0\ (1). 

The original value of the angle SPQ was -^-6, and the increase due to 

displacement is i{Pi — Pi) cot ^ - ^ (o-j - Ci), 



ON THE STABILITY OF THE MOTION OF SATURN 8 RINGS. 



313 



30 that we may write the values of sin SPQ and cos SPQ, 

Gin SFQ = cos e {I +i{p,-p,)-i {a-,- a,) ta,n0} (2), 

cos SPQ = am e {I -i(p,-p,)coVd + i (a-,- a-,) cot 6} (3). 



If we assume the masses of P and Q each equal to - R, where P is the 

mass of the ring, and p, the number of satellites of which it is composed, the 
accelerating effect of the radial force on P is 

li}22^ = l--«_^{l_(p. + p,)_i(p._p,)eof^-iK-.T.)cot3}...(4), 

and the tangential force 

I j^sinSPQ li^COS^-. , \ / + ^ , l x mi /r:\ 

]1^ PQ ^^H^i^I^-^/^^-f/^^-l^'-^Olcot^ + itan^)} (5). 

1 L — l 
The normal force is -R ^ . , \. 
p. 8 sm^ 6 

5. Let us substitute for p, or and { their values expressed in a series of 
sines and cosines of multiples of 5, the terms involving ms being 

Pi = A cos {ms + a), pi = A cos (ms + a + 20), 

o-, = -Bsin(m5 + ^), cr. = B sin {7}is-\-fi + 20), 

C, = C cos (ms + y), C2 = Ccos {ms + y + 26). 

The radial force now becomes 

1 — ^ cos {ms + a) ( 1 + cos 2m0) + A sin {ms + a) sin 2md i 

+ ^A cos {ms + a) (1 - cos 2m6) cot' ^ - ^^ sin (t/i^ + a) sin 2ni6 cot" 6 \ (6). 
+^B sin {ms + ft) {1 -cos 2m^) cot ^-^5cos(??i5 + /8) siii2w^cot^. 

The radial component of the attraction of a corresponding particle on the 
other side of P may be found by changing the sign of 6. Adding the two 
together, we have for the effect of the pair 

- ^-^ — ^ {1 — ^ COS {ms + a) (2 cos" md — sin' md cot' 6) 

- B cos {ms + 13) ^ sin 2m6 cot 6] 



I_i2_ 
/x 4 sin ^ 



(?)• 



314 



Let us put 



ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 

sin* mO cos'' 6 cos' m6\ 



K=t 



f^va. 2m6 cos 
\ 4sin*^ 

/sin" md cos' 6 

sin' mt 
2 sin^ 

1 







+ i 



sin'?n^ 



2sin^ 



(8)^: 



where the summation extends to all the sateUites on the same side of F, 
that is, every value of 6 of the form - tt, where x is a whole number less 



than 



The radial force may now be written 



P = ~R {K+ LA cos ims + a) - MB cos {'tm + ^)} 



(9). 



* Tlie following values of several quantities which enter into these investigations are calculated for a 
ring of 36 satellites. 

A' =24-5. 





^ sin- md cos- $ ^ cos^ md ^ 




if 


.V 




sinS d sin e 








m= 


43 - 43 










ni= 1 


32 32 -16 




16 


37 


m= 2 


107 28 26 




25 


115 


m — 3 


212 25 81 




28 


221 


;u= 4 


401 24 177 




32 


411 


vi= 9 


975 20 468 




30 


986 


/ft- 18 


1569 18 767 







1582 


r gi-eat, 


- Z - -5259 when m -- 
= '4342 „ TO = 
= -3287 „ m = 


"3' 







ON THE STABILITY OF THE MOTION OF SATURN 8 RINGS. 



315 



The tangential force may be calculated in the same way, it is 

T=- R{MAam(iiis-\-a) + NBsm(7ns + IB)} (10). 



The normal force is 



Z= -^-RJC cos (ms + y) (11). 



G. We have found the expressions for the forces which act upon each 
member of a system of equal satellites which originally formed a uniform ring, 
but are now aflfected with displacements depending on circular functions. If 
these displacements can be propagated round the ring in the form of waves 

with the velocity — , the quantities a, y8, and y will depend on t, and the 

complete expressions will be 

p = ^ cos (ms + nt-\- a) ' 

a = Bam(ms + nt+^) ■ (12). 

^ = Ccos (ms + nt + y). 
Let us find in what cases expressions such as these will be true, and 
what will be the result when they are not true. 

Let the position of a satellite at any time be determined by the values 
of r, (j), and C, where r is the radius vector reduced to the plane of reference, 
<t> the angle of position measured on that plane, and ^ the distance from it. 
The equations of motion will be 



[dtj df ^ r-^-^ 



dr d4 d^_^ 
^Tt dt ^"^ df~ 

d^ 

df' 



1^ 



.(13). 



If we substitute the value of ^ in the third equation and remember that r 
is nearly = 1 , we find 






(14). 



As this expression is necessarily positive, the value of n' is always real, 
and the disturbances normal to the plane of the ring can always be propa- 



31G ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 

gated as waves, and therefore can never be the cause of instability. We 
therefore confine our attention to the motion in the plane of the ring as 
deduced from the two former equations. 

Putting r = 1 4- /) and (f> = <ot + s + a; and omitting powers and products of 
p, cr and their differential coeflScients, 



''+'">+2-t-t='^-2«''+^ 



-l+§=^ 



(15). 



Substituting the values of p and cr as given above, these equations become 
oi'-S-- RK+ U- -]-2S--EL + 7f)A cos (ttis + nt + a) 

+ (2(071 + -RM)B COS (ins + nt + ^) = ...(16), 

H' 

(2(071 + - EM) A sin (ins + nt + a) + (if +-RN)Bam(7ns + nt-\-^) = 0.... (17). 
p p 

Putting for (ins + nt) any two diflferent values, we find from the second 
equation (17) 

a=)8 (18), 

and (2(on + -E]\f)A + (n'+-EN)B = (19), 

and from the first (16) ((o' + 2S -- EL + iv) A + (2(on + - EM) B = (20), 

and (o'-S--EK=0 (21). 

p 

Eliminating A and B from these equations, we get 
n'-{S(o'-2S + -E(L-N)}n^ 

-'4(o-EMn + ((o' + 2S--EL)-EN--,E'M' = (22), 

a biquadratic equation to determine n. 

For every real value of n there are terms in the expressions for p and o- 

of the form 

A cos (nis + nt + a). 



ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 317 

For every pure impossible root of the form ±7 — In' there are terms of 
the forms 

^e^^'cos (ms + a). 

Although the negative exponential coefficient indicates a continually diminlshmg 
displacement which is consistent with stability, the positive value which neces- 
sarily accompanies it indicates a continually increasing disturbance, which would 
completely derange the system in course of time. 

For every mixed root of the form ±n/ — In' + n, there are terms of the form 

.46*"'' cos {ms + nt + a). 

If we take the positive exponential, we have a series of m waves travelling 

with velocity — and increasing in amplitude with the coefficient e"^"'. The 

negative exponential gives us a series of m waves gradually dying away, but 
the negative exponential cannot exist without the possibility of the positive one 
having a finite coefficient, so that it is necessary for the stability of the motion 
that the four values of n be all real, and none of them either impossible 
quantities or the sums of possible and impossible quantities. 

We have therefore to determine the relations among the quantities K, L, 
M, N, R, S, that the equation 

n'-lS+^RidK+L-N)]?^ 

'-4<o-RMn + {SS+ - R (K-L)} - RN- \ R'M'^ U=0 

may have four real roots. 

7. In the first place, U is positive, when tz is a large enough quantity, 
whether positive or negative. 

It is also positive when 7i=;0, provided S be large, as it must be, com- 
pared with - RL, - RM and - RN. 

If we can now find a positive and a negative value of n for which U 
is negative, there must be four real values of n for which U=0, and the four 
roots will be real. 



318 ON THE STABILITY OF THE MOTION OF SATURN S RINGS. 

Now if we put n= ±J^JS, 

U= -^S' + l -R{7N±ij2M-L-dK) S+ \r{KN-LN^M% 

which is negative if >S be large compared to R. 

So that a ring of satellites can always be rendered stable by increasing 
the mass of the central body and the angular velocity of the ring. 

The values of L, M, and N depend on m, the number of undulations in 

the ring. When m = ^, the values of L and N will be at their maximum 

and M=0. If we determine the relation between S and R in this case so 
that the system may be stable, the stability of the system for every other 
displacement will be secured. 

8. To find the mass which must be given to the central body in order 
that a ring of satellites may permanently revolve round it. 

We have seen that when the attraction of the central body is sufficiently 
great compared with the forces arising from the mutual action of the satellites, 
a permanent ring is possible. Now the forces between the satellites depend on 
the manner in which the displacement of each satellite takes place. The con- 
ception of a perfectly arbitrary displacement of all the satellites may be rendered 
manageable by separating it into a number of partial displacements depending 
on periodic functions. The motions arising from these small displacements will 
take place independently, so that we have to consider only one at a time. 

Of all these displacements, that which produces the greatest disturbing 
forces is that in w^hich consecutive satellites are oppositely displaced, that is, 

when m = -, for then the nearest satellites are displaced so as to increase as 

z 

much as possible the effects of the displacement of the satellite between them. 
If we make /x a large quantity, we shall have 

2™^<^ = e;(l + 3-' + 5- + &c.) = ^.(l-0518). 
sm^ n' ^ TT 

M=0, N=2L, J5r very small. 



IT 



ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 319 

Let - RL = X, then the equation of motion will be 
/A 

n*-{S-x)n' + 2x{'iS-x)=U=0 (23). 

The conditions of tliis equation having real roots are 

S>x (24), 

(S-xY>^x{'iS-x) (25). 

The last condition gives the equation 

6:'-26*Sx + 9ar>0, 

whence S>2Q-U2x, or>S<0-351a; (26). 

The last solution is inadmissible because S must be greater than x, so that 

the true condition is »S>25*649a:, 

> 25-649 i 72^3 -5259, 

/X IT 

S>-ASd2im'R (27). 

So that if there were 100 satellites in the ring, then 

5>4352i2 
is the condition which must be fulfilled in order that the motion arising from 
every conceivable displacement may be periodic. 

If this condition be not fulfilled, and if S be not sufiadent to render the 
motion perfectly stable, then although the motion depending upon long undu- 
lations may remain stable, the short undulations wiill increase in amplitude till 
some of the neighbouring satellites are brought into collision. 

9. To determine the nature of the motion when the system of satellites 
is of small mass compared with the central body. 

The equation for the determination of n is 

^ /x ^ /x 

+ {Zoy-- R{2K+L)]~ RN -\R'M'=^0 (28). 

F' r" r" 

When R is very small we may approximate to the values of n by assuming 
that two of them are nearly ± co, and that the other two are small. 



320 ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 

If we put n= ±(0, 



dU 

dn 



= ±2g>' + &c. 



Therefore the corrected values of n are 



n^±{<o + ^R(2K + L-.m)} + ^RM. 



(29). 



The small values of n are nearly ±/3-i2iV^: correcting them in the 
way, we find the approximate values 

n=±./3^EN^2~RM 



same 



(30). 



The four values of n are therefore 

1 



^1= -<o-^-E{2K+L^iM-4N) 



RN- — RM 

fXCt) 



^z=+J^-RN- — RM 



(31), 



^4= +o>+^--R(2K+L + iM-4N) 

and the complete expression for p, so far as it depends on terms containing ms, 
is therefore P = A, cos {ms + n^t + a^)-\-A^ cos (ws + n^t + c^) 

+ A^co&(ms + nJ, + a^-{-A^coB{ms-\-nJ^ + a^) (32), 

and there will be other systems, of four terms each, for every value of m in 
the expansion of the original disturbance. 

We are now able to determine the value of o- from equations (12), (20), by 
putting /8 = a, and 



2<an + - RM 



5= — 



(33). 



n' + -RN 



ON THE STABILITY OF THE MOTION OF SATURN'S RINGS. 321 

So that for every term of p of the form 

p = Acos (ms -{-111 + a) (34), 

there is a corresponding term in a, 



2w7i + - RM 



7t' + -RN 



A sin {ms-¥7it + a) (35). 



10. Let us now fix our attention on the motion of a single satellite, 
and determine its motion by tracing the changes of p and a- while t varies 
and 5 is constant, and equal to the value of s corresponding to the satellite 
in question. 

We must recollect that p and a- are measured outwards and forwards from 
an imaginary point revolving at distance 1 and velocity o, so that the motions 
we consider are not the absolute motions of the satellite, but its motions 
relative to a point fixed in a revolving plane. This being understood, we may 
describe the motion as elliptic, the major axis being in the tangential direc- 
tion, and the ratio of the axes being nearly 2 ^ , which is nearly 2 for n, and n, 
and is very large for n^ and n^. 

The time of revolution is — , or if we take a revolution of the ring as 

the unit of time, the time of a revolution of the satellite about its mean 

... . it) 
position IS - . 



The direction of revolution of the satellite about its mean position is in 
every case opposite to that of the motion of the ring. 

11. The absolute motion of a satellite may be found from its motion 
relative to the ring by writing 

r=l+p = l+^cos {ms + nt + a), 
d = (ot + s-{-<T = (ot + s-2 -Asm{ms-\-nt-\-a). 



322 ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 

When n is nearly equal to ±(0, the motion of each satellite in space is 
nearly elliptic. The eccentricity is A, the longitude at epoch s, and the longi- 
tude when at the greatest distance from Saturn is for the negative value n^ 

- — R{2K+L-iM-4N)t + {m+l)s + a, 
and for the positive value n^ 

- — R{2K+L + 4M^4.N)t-{m+l)s-a. 

We must recollect that in all cases the quantity within brackets is negative, 
so that the major axis of the ellipse travels forwards in both cases. The chief 
difference between the two cases lies in the arrangement of the major axes of 
the ellipses of the different satellites. In the first case as we pass from one 
satellite to the next in front the axes of the two ellipses lie in the same 
order. In the second case the particle in front has its major axis behind that 
of the other. In the cases in which n is small the radius vector of each 
satellite increases and diminishes during a periodic time of several revolutions. 
This gives rise to an inequality, in which the tangential displacement far exceeds 
the radial, as in the case of the annual equation of the Moon. 

12. Let us next examine the condition of the ring of satellites at a given 
instant. We must therefore fix on a particular value of t and trace the changes 
of p and <r for different values of s. 

From the expression for p we learn that the satellites form a wavy line, 
which is furthest from the centre when (ms + nt + a) is a multiple of 27r, and 
nearest to the centre for intermediate values. 

From the expression for cr we learn that the satellites are sometimes in 
advance and sometimes in the rear of their mean position, so that there are 
places where the satellites are crowded together, and others where they are 
drawn asunder. When n is positive, ^ is of the opposite sign to A, and the 
crowding of the satellites takes place when they are furthest from the centre. 
When n is negative, the satellites are separated most when furthest from the 
centre, and crowded together when they approach it. 

The form of the ring at any instant is therefore that of a string of beads 
forming a re-entering curve, nearly circular, but with a small variation of distance 



ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 323 

from the centre recurring m times, and forming m regular waves of trans- 
vei-se displacement at equal intervals round the circle. Besides these, there are 
waves of condensation and rarefaction, the effect of longitudinal displacement. 
When n is positive the points of greatest distance from the centre are points 
of greatest condensation, and when n is negative they are points of greatest 
rarefaction. 

13. We have next to determine the velocity with which these waves of 
disturbance are propagated round the ring. We fixed our attention on a par- 
ticular satellite by making s constant, and on a particular instant by making t 
constant, and thus we determined the motion of a satellite and the form of the 
ring. We must now fix our attention on a phase of the motion, and this we 
do by making p or a- constant. This implies 

ms + nt + a = constant, 

ds _ n 
dt~ m* 

So that the particular phase of the disturbance travels round the ring with an 

angular velocity = relative to the ring itself. Now the ring is revolving 

in space with the velocity w, so that the angular velocity of the wave in space is 

tj- = w (36). 

m 

Thus each satellite moves in an ellipse, while the general aspect of the 
ring is that of a curve of m waves revolving with velocity ct. This, however, 
is only the part of the whole motion, which depends on a single term of the 
solution. In order to understand the general solution we must shew how to 
determine the whole motion from the state of the ring at a given instant. 

14. Given the position and motion of every satellite at any one time, to 
calculate the position and motion of every satellite at any other time, provided 
that the condition of stability is fulfilled. 

The position of any satellite may be denoted by the values of p and cr for 
that satellite, and its velocity and direction of motion are then indicated by the 

values of -r and -y- at the g:iven instant. 
dt at 



324 ON THE STABILITY OF THE MOTION OF SATURN S RINGS. 

These four quantities may have for each satellite any four arbitrary values, 
as the position and motion of each satellite are independent of the rest, at the 
beginning of the motion. 

Each of these quantities is therefore a perfectly arbitrary ftmction of s, the 
mean angular position of the satellite in the ring. 

But any function of s from s = to s = 27r, however arbitrary or discontinuous, 
can be expanded in a series of terms of the form A cos (5 + a) + A' cos (2s + a') + &c. 
See § 3. 

Let each of the four quantities p, -^ , a, -j- he expressed in terms of such 

a series, and let the terms in each involving ms be 

p = Ecoa{'ms + e) (37), 

^^=Fcos(ins+f) (38). 

<T =G cos (ms+g) (39), 

^ = Hco3{ms + h) (40). 

These are the parts of the values of each of the four quantities which are 
capable of being expressed in the form of periodic fimctions of ms. It is 
evident that the eight quantities E, F, G, H, e, f, g, h, are all independent and 
arbitrary. 

The next operation is to tind the values of X, M, N, belonging to disturb- 
ances in the ring whose index is m [see equation (8)], to introduce these 
values into equation (28), and to determine the four values of n, (ti,, tIj, 1I3, n^). 

This being done, the expression for p is that given in equation (32), which 
contains eight arbitrary quantities (A,, A^, A3, At, «„ a^, a^, aj. 

Giving t its original value in this expression, and equating it to Eco3{7m-\-e), 
we get an equation which is equivalent to two. For, putting 7ns = 0, we have 
^1 cos Oi + .^2 cos a, + -^3 cos a, + ^^ cos a^ = -E' cos e (41). 

And putting ms= , we have another equation 

-4i sin Oi + ^j sin aj + ^3 sin 03 + ^< sin a^ = ^ sin e (42). 



ON THE STABILITY OF THE MOTION OF SATUBN's RINGS. 325 

Differentiating (32) with respect to t, we get two other equations 

- A^n^ Bina-kc.-F cos/ (43), 

Aji^ cos a + &c.=F sin/ (44 ). 

Bearing in mind that B„ B^, &c. are connected with A„ A^, &c. by equa- 
tion (33), and that B is therefore proportional to A, we may write B = A^, 
where 

2o)n + - RM 

P ^ 
P= 7 

H' 
^ being thus a fiinction of n and a known quantity. 

The value of <r then becomes at the epoch 

<r = ^i)8i sin (m5 4- Oi) -I- &c. = Gcoa('ms-\-g), 
from which we obtain the two equations 

^^1 sin Oi -I- &c. = 6^ cos g (45), 

^^iC0Sai + &c. = —Geing (46). 

Differentiating with respect to t, we get the remaining equations 

A^jij^ cos Oj + &c. = ^ cos A (47), 

^^iniSinai-l-&c. = iZ'sinA (48). 

We have thus found eight equations to determine the eight quantities 
^1, &c. and Oi, &c. To solve them, we may take the four in which -^iCosoi, 
&c. occur, and treat them as simple equations, so as to find ^iCosoj, &c. Then 
taking those in which ^isinoi, &c. occur, and determining the values of those 
quantities, we can easily deduce the value of A^ and a,, &c. from these. 

We now know the amplitude and phase of each of the four waves whose 
index is m. All other systems of waves belonging to any other index must 
be treated in the same way, and since the original disturbance, however arbitrary, 
can be broken up into periodic functions of the form of equations (37 — 40), 
our solution is perfectly general, and applicable to every possible disturbance of 
a ring fulfilling the condition of stability (27). 



326 ON THE STABILITY OF THE MOTION OF SATURN S RINGS. 

15. We come next to consider the effect of an external disturbing force, 
due either to the irregularities of the planet, the attraction of satellites, or 
the motion of waves in other rings. 

All disturbing forces of this kind may be expressed in series of which the 
general term is 

A cos {vt + ms + a), 

where v is an angular velocity and m a whole number. 

Let P cos {ins + vt +p) be the central part of the force, acting inwards, and 
Q sin (ms + vt + q) the tangential part, acting forwards. Let p = A cos {tus + vt + a) 
and a- = Bsm (ms + vt-]- fi), be the terms of p and a which depend on the 
external disturbing force. These will simply be added to the terms depending 
on the original disturbance which we have already investigated, so that the 
complete expressions for p and <t will be as general as before. In consequence 
of the additional forces and displacements, we must add to equations (16) and 
(17), respectively, the following terms: 

{Zar --R (2K+ L) + v"] A cos (m^-{-vt-\- a) 

+ (2q)V -\- - RM) B COS (ms + vt + f3)-P cos (ms + vt-hp) = (49). 

(2a)i; 4- - EM) A sin (ms + vt + a) 

+ (v" + - EN) B Bm(ms + vt + fi)-¥Q sin (ms + vt + q) = (50). 

Making 7ns + vt = in the first equation and - in the second, 

{S(o'-- E (2K+L) + if} A cos a + (2(ov + -E3f) B cos fi-P coap = (51). 

(2a>v + - EM) A cosa + (v' + - EN) B COB fi + Qcosq = (52). 



Then if we put 
U' = v'-{oj' + -E(2K+L-N)}v'-A-EMv 



+ {Sa>'--E(2K+L)}-EN-\E'M' (53), 



ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 327 

we shall find the value of A cos a and B coa fi ; 

v' + -RN 2cov-i-~RM 
A cosa = ft; P coa p + t4 Qcoaq (54). 

2(ov 4- - RM y' + 3<o'--R {K+ L) 
Bcoafi= j^ Pcoap jp Qcoaq (55). 

Substituting sines for cosines in equations (51), (52), we may find the 
values of A sin a and B sin ^. 

Now U* is precisely the same function of v that Z7 is of ?i, so that if u 
coincides with one of the four values of n, U' will vanish, the coefiicients A 
and B will become infinite, and the ring will be destroyed. The disturbing 
force is supposed to arise from a revolving body, or an undulation of any kind 

which has an angular velocity relatively to the ring, and therefore an 

absolute angular velocity = w . 

If then the absolute angular velocity of the disturbing body is exactly or 
nearly equal to the absolute angular velocity of any of the free waves of the 
ring, that wave will increase till the ring be destroyed. 

The velocities of the free waves are nearly 

l+i\ a> + i /s-i^.V, o>-- /s-i^iV^, and 0) fl-i) (56). 

When the angular velocity of the disturbing body is greater than that of 
the first wave, between those of the second and third, or less than that of 
the fourth, U' is positive. When it is between the first and second, or between 
the third and fourth, U' is negative. 

Let us now simplify our conception of the disturbance by attending to the 
central force only, and let us put ^ = 0, so that P is a maximum when ms + vt 
is a multiple of 27r. We find in this case a = 0, and /8 = 0. Also 

if+^- RN 
^=—^P (57), 

2cjv + -RM 
B= ^. P (58). 



328 ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 

When U' is positive, A will be of the same sign as P, that is, the parts 
of the ring wlU be furthest from the centre where the disturbing force towards 
the centre is greatest. When U' is negative, the contrary will be the case. 

When V is positive, B will be of the opposite sign to A, and the parts 
of the ring furthest from the centre will be most crowded. When v is negative, 
the contrary will be the case. 

Let us now attend only to the tangential force, and let us put ^' = 0. We 
find in this case also a = 0, )3 = 0, 

2(ov+-RM 

^= — tr — ^ (^^)' 

B= ^. Q (60). 

The tangential displacement is here in the same or in the opposite direc- 
tion to the tangential force, according as £/"' is negative or positive. The 
crowding of sateUites is at the points farthest from or nearest to Saturn 
according as -y is positive or negative. 

16. The effect of any disturbing force is to be determined in the following 
manner. The disturbing force, whether radial or tangential, acting on the ring 
may be conceived to vary from one satellite to another, and to be different at 
different times. It is therefore a perfectly arbitrary function of s and t. 

Let Fourier's method be applied to the general disturbing force so as to 
divide it up into terms depending on periodic functions of s, so that each term 
is of the form F (t) cos {ms + a), where the function of i is still perfectly arbitrary. 

But it appears from the general theory of the permanent motions of the 
heavenly bodies that they may all be expressed by periodic functions of t 
arranged in series. Let vt be the argument of one of these terms, then the 
corresponding term of the disturbance will be of the form 

P cos (ttis + vt + a). 

This term of the disturbing force indicates an alternately positive and 
negative action, disposed in m waves round the ring, completing its period 



ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 329 

relatively to eaxih particle in the time — , and travelling as a wave among 

the particles with an angular velocity , the angular velocity relative to fixed 

space being of course oj — - . The whole disturbing force may be split up into 
terms of this kind. 

17. Each of these elementary disturbances will produce its own wave in 
the ring, independent of those which belong to the ring itself. This new wave, 
due to external disturbance, and following different laws from the natural waves 
of the rincy, is called the farced wave. The angular velocity of the forced wave 
is the same as that of the disturbing force, and its maxima and minima coin- 
cide with those of the force, but the extent of the disturbance and its direction 
depend on the comparative velocities of the forded wave and the four natural 
waves. 

When the velocity of the forced wave lies between the velocities of the 
two middle free waves, or is greater than that of the swiftest, or less than 
that of the slowest, then the radial displacement due to a radial disturbing 
force is in the same direction as the force, but the tangential displacement 
due to a tangential disturbing force is in the opposite direction to the force. 

The radial force therefore in this case produces a positive forced wave, and 
the tangential force a negative forced ivave. 

When the velocity of the forced wave is either between the velocities of 
the first and second free waves, or between those of the third and fourth, then 
the radial disturbance produces a forced wave in the contrary direction to that 
in which it acts, or a negative wave, and the tangential force produces a positive 
wave. 

The coefficient of the forced wave changes sign whenever its velocity passes 
through the value of any of the velocities of the free waves, but it does so 
by becoming infinite, and not by vanishing, so that when the angular velocity 
very nearly coincides with that of a free wave, the forced wave becomes very 
great, and if the velocity of the disturbing force were made exactly equal t-o 
that of a free wave, the coefficient of the forced wave would become infinite. 
In such a case we should have to readjust our approximations, and to find 
whether such a coincidence might involve a physical impossibility. 



330 ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 

The forced wave which we have just investigated is that which would main- 
tain itself in the ring, supposing that it had been set agoing at the commence- 
ment of the motion. It is in fact the form of dynamical equiUbrium of the 
ring under the influence of the given forces. In order to find the actual motion 
of the ring we must combine this forced wave with all the free waves, which 
go on independently of it, and in this way the solution of the problem becomes 
perfectly complete, and we can determine the whole motion under any given 
initial circumstances, as we did in the case where no disturbing force acted. 

For instance, if the ring were perfectly uniform and circular at the instant 
when the disturbing force began to act, we should have to combine with the 
constant forced wave a system of four free waves so disposed, that at the given 
epoch, the displacements due to them should exactly neutralize those due to the 
forced wave. By the combined effect of these four free waves and the forced 
one the whole motion of the ring would be accounted for, beginning from its 
undisturbed state. 

The disturbances which are of most importance in the theory of Saturn's 
rings are those which are produced in one ring by the action of attractive 
forces arising from waves belonging to another ring. 

The effect of this kind of action is to produce in each ring, besides its 
own four free waves, four forced waves corresponding to the free waves of the 
other ring. There will thus be eight waves in each ring, and the corresponding 
waves in the two rings will act and react on each other, so that, strictly speak- 
ing, every one of the waves will be in some measure a forced wave, although 
the system of eight waves will be the free motion of the two rings taken 
together. The theory of the mutual disturbance and combined motion of two 
concentric rings of satellites requires special consideration. 

18. On the motion of a ring of satellites when the conditions of stability 
are not fulfilled. 

We have hitherto been occupied with the case of a ring of satellites, the 
stability of which was ensured by the smaUness of mass of the satellites com- 
pared with that of the central body. We have seen that the statically unstable 
condition of each satellite between its two immediate neighbours may be com- 
pensated by the dynamical effect of its revolution round the planet, and a planet 
of sufiicient mass can not only direct the motion of such satellites round its 



ON THE STABILITY OF THE MOTION OF SATURN S RINGS. 331 

own body, but can likewise exercise an influence over their relations to each 
other, so as to overrule their natural tendency to crowd together, and distribute 
and preserve them in the form of a ring. 

We have traced the motion of each satellite, the general shape of the 
disturbed ring, and the motion of the various waves of disturbance round the 
ring, and determined the laws both of the natural or free waves of the ring, 
and of the forced waves, due to extraneous disturbing forces. 

We have now to consider the cases in which such a permanent motion of 
the ring is impossible, and to determine the mode in which a ring, originally 
regular, will break up, in the different cases of instability. 

The equation from which we deduce the conditions of stability is — 
U = n'-i(o' + -E(2K+L-N)\n'-4:(o-EMn 

+ hco'--R{2K + L)\-RN -\r'M' = 0. 

The quantity, which, in the critical cases, determines the nature of the 
roots of this equation, is N. The quantity M in the third term is always 
small compared with L and N when m is large, that is, in the case of the 
dangerous short waves. We may therefore begin our study of the critical cases 
by leaving out the third term. The equation then becomes a quadratic in n\ 
and in order that all the values of n may be real, both values of n' must be 
real and positive. 

The condition of the values of n^ being real is 

oj* + co'-R{AK + 2L-UN) + \b'{2K+L-\-NY>0 (61), 

which shews that ay must either be about 14 times at least smaller, or about 14 
times at least greater, than quantities like - RN. 



That both values of if may be positive, we must have 
co' + -R{2K + L-N)>0 



i3co''--R(2K-^L)\-RN>0 



(62). 



332 ON THE STABILITY OF THE MOTION OF SATURN S EINGS. 

We must therefore take the larger value 6£ oi\ and also add the condition 
that N be positive. 

RN 

We may therefore state roughly, that, to ensure stability, , the coefficient 

of tangential attraction, must lie between zero and -^oi\ If the quantity be 
negative, the two small values of n will become _pwre impossible quantities. If 
it exceed ^oi\ all the values of n will take the form of mixed impossible 
quantities. 

If we write x for - RN, and omit the other disturbing forces, the equation 

becomes U=n*-{(o'-x)n' + Sco'x = (63), 

whence n' = ^{co'-x)±^^/<o*-U(o'x + x' (64). 

If X be small, two of the values of n are nearly ±<o, and the others are 
small quantities, real when x is positive and impossible when x is negative. 

2 

If x be greater than {7-^IS)ar, or ^ nearly, the term under the radical 
becomes negative, and the value of ?i becomes 

n= ±^^fjT2^ + o}'-x±^/^-^'Jl2co'x-ajr + x (65), 

where one of the terms is a real quantity, and the other impossible. Every 
solution may be put under the form 

n=p±J^^q (66), 

where ry = for the case of stability, p = for the pure impossible roots, and p 
and q finite for the mixed roots. 

Let us now adopt this general solution of the equation for n, and determine 
its mechanical significance by substituting for the impossible circular functions 
their equivalent real exponential functions. 

Substituting the general value of n in equations (34), (35), 

p = A[cos {ms +(p + 'J^^q)t + a} + cos {ms + ip- J -lq)t + a}] ... (67), 



^^_^MP+±zlAsm{,ns + (p + ^^q)t + a} ] 
(p + J-lqf + x 

_^MEpdIi^sm{ms+(p-sr^lq)t + a} \ 
{p-'J -IqY + x J 



ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 333 

Introducing the exponential notation, these values become 

p = A(^^ + €-''')co3(ms-{-pt + a) (69), 

W r 2) (^' + r/ + x) (€«' + €-«') sin (771,5 +j9« + a) 1 

We have now obtained a solution free from impossible quantities, and applicable 
to every case. 

When ^ = 0, the case becomes that of real roots, which we have already 
discussed. When p = 0, we have the case of pure impossible roots arising from 
the negative values of if. The solutions corresponding to these roots are 

/3 = ^ (e«' + €-«') cos (m5 + a) (71). 

o-=-^r^^^(€''-e-^0cos(m5 + a) (72). 

The part of the coefficient depending on e"'' diminishes indefinitely as the 
time increases, and produces no marked effect. The other part, depending on 
€^', increases in a geometrical proportion as the time increases arithmetically, and 
so breaks up the ring. In the case of x being a small negative quantity, q' is 
nearly 3x, so that the coefficient of cr becomes 

It appears therefore that the motion of each particle is either outwards and 
backwards or inwards and forwards, but that the tangential part of the motion 
greatly exceeds the normal part. 

It may seem paradoxical that a tangential force, acting towards a position 
of equilibrium, should produce instability, while a small tangential force from that 
position ensures stability, but it is easy to trace the destructive tendency of 
this apparently conservative force. 

Suppose a particle slightly in front of a crowded part of the ring, then 
if X is negative there will be a tangential force pushing it fonvards, and this 
force will cause its distance from the planet to increase, its angular velocity U> 
diminish, and the particle itself to fall back on the crowded part, thereby 
increasing the irregularity of the ring, till the whole ring is broken up. In 
the same way it may be shewn that a particle hehiiid a crowded part will be 
pushed into it. The only force which could preserve the ring from the effect 



334 ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 

of tills action, is one which would prevent the particle from receding from the 
planet under the influence of the tangential force, or at least prevent the dimi- 
nution of angular velocity. The transversal force of attraction of the ring is of 
this kind, and acts in the right direction, but it can never be of sufficient magni- 
tude to have the required effect. In fact the thing to be done is to render the 
last term of the equation in w positive when N is negative, which requires 

fX 

and this condition is quite inconsistent with any constitution of the ring which 
fiilfils the other condition of stability which we shall arrive at presently. 

We may observe that the waves belonging to the two real values of n, 
±(D, must be conceived to be travelling round the ring during the whole time 
of its breaking up, and conducting themselves like ordinary waves, till the 
excessive irregularities of the ring become inconsistent with their uniform propa- 
gation. 

The irregularities which depend on the exponential solutions do not travel 
round the ring by propagation among the sateUites, but remain among the same 
satellites which first began to move irregularly. 

We have seen the fate of the ring when x is negative. When x is small 
we have two small and two large values of n, which indicate regular waves, 
as we have already shewn. As x increases, the small values of n increase, and 
the large values diminish, till they meet and form a pair of positive and a 
pair of negative equal roots, having values nearly +"68w. When x becomes 
greater than about -^(o", then all the values of n become impossible, of the 
form ^j-F-n/ — Ig", q being small when x first begins to exceed its limits, and p 
being nearly + '6S(o. 

The values of p and cr indicate periodic inequalities having the period — , 

but increasing in amplitude at a rate depending on the exponential e''. At the 
beginning of the motion the oscillations of the particles are in eUipses as in the 
case of stability, having the ratio of the axes about 1 in the normal direction 
to 3 in the tangential direction. As the motion continues, these ellipses increase 
in magnitude, and another motion depending on the second term of cr is com- 
bined with the former, so as to increase the ellipticity of the oscillations and to 



ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 335 

turn the major axis into an inclined position, so that its fore end points a little 
inwards, and its hinder end a little outwards. The oscillations of each particle 
round its mean position are therefore in ellipses, of which both axes increase 
continually while the eccentricity increases, and the major axis becomes sUghtly 
inclined to the tangent, and this goes on till the ring is destroyed. In the 
mean time the irregularities of the ring do not remain among the same set of 
particles as in the former case, but travel round the ring^ with a relative angular 

velocity - ^^ Of these waves there are four, two travelling forwards among the 

satellites, and two travelling backwards. One of each of these pairs depends 
on a negative value of q, and consists of a wave whose amplitude continually 
decreases. The other depends on a positive value of q, and is the destructive 
wave whose character we have just described. 

19. We have taken the case of a ring composed of equal satellites, as 
that with which we may compare other cases in which the ring is constructed 
of loose materials diiferently arranged. 

In the first place let us consider what will be the conditions of a ring 
composed of satellites of unequal mass. We shall find that the motion is of 
the same kind as when the satellites are equal. 

For by arranging the satellites so that the smaller satellites are closer 
together than the larger ones, we may form a ring which will revolve uni- 
formly about Saturn, the resultant force on each satellite being just sufficient 
to keep it in its orbit. 

To determine the stability of this kind of motion, we must calculate the 
disturbing forces due to any given displacement of the ring. This calculation 
will be more complicated than in the former case, but will lead to results of 
the same general character. Placing these forces in the equations of motion, 
we shall find a solution of the same general character as in the former case, 
only instead of regular waves of displacement travelling round the ring, each 
wave will be split and reflected when it comes to irregularities in the chain of 
satellites. But if the condition of stability for every kind of wave be fulfilled, 
the motion of each satellite will consist of small oscillations about its position 
of dynamical equilibrium, and thus, on the whole, the ring will of itself assume 
the arrangement necessary for the continuance of its motion, if it be originally 
in a state not very different from that of equilibrium. 



336 ON THE STABILITY OF THE MOTION OF SATURN'S RINGS. 

20. We now pass to the case of a ring of an entirely different construc- 
tion. It is possible to conceive of a quantity of matter, either solid or liquid, 
not collected into a continuous mass, but scattered thinly over a great extent 
of space, and having its motion regulated by the gravitation of its parts to 
each other, or towards some dominant body. A shower of rain, hail, or cinders 
is a familiar illustration of a number of unconnected particles in motion; the 
visible stars, the milky way, and the resolved nebula?, give us instances of a 
similar scattering of bodies on a larger scale. In the terrestrial instances we 
see the motion plainly, but it is governed by the attraction of the earth, and 
retarded by the resistance of the air, so that the mutual attraction of the 
parts is completely masked. In the celestial cases the distances are so enor- 
mous, and the time during which they have been observed so short, that we 
can perceive no motion at all. StiU we are perfectly able to conceive of a 
collection of particles of small size compared with the distances between them, 
acting upon one another only by the attraction of gravitation, and revolving 
round a central body. The average density of such a system may be smaller 
than that of the rarest gas, while the particles themselves may be of great 
density ; and the appearance from a distance will be that of a cloud of vapour, 
with this difference, that as the space between the particles is empty, the rays 
of light will pass through the system without being refracted, as they would 
have been if the system had been gaseous. 

Such a system will have an average density which may be greater in some 
places than others. The resultant attraction wiU be towards places of greater 
average density, and thus the density of those places wiU be increased so as 
to increase the irregularities of density. The system will therefore be statically 
unstable, and nothing but motion of some kind can prevent the particles from 
forming agglomerations, and these uniting, till all are reduced to one solid 
mass. 

We have already seen how dynamical stability can exist where there is 
statical instability in the case of a row of particles revolving round a central 
body. Let us now conceive a cloud of particles forming a ring of nearly uni- 
form density revolving about a central body. There will be a primary effect of 
inequalities in density tending to draw particles towards the denser parts of the 
ring, and this will ehcit a secondary effect, due to the motion of revolution, 
tending in the contrary direction, so as to restore the rings to uniformity. The 



ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 337 

relative magnitude of these two opposing forces determines the destruction or 
preservation of the ring. 

To calculate these effects we must begin with the statical problem : — To 
determine the forces arising from the given displacements of the ring. 

The longitudinal force arising from longitudinal displacements is that which 
has most effect in determining the stability of the ring. In order to estimate ita 
limiting value we shall solve a problem of a simpler form. 

21. An infinite mass, originally of uniform density Tc, has its particles 
displaced by a quantity f parallel to the axis of x, so that ^ = AcQ^mx, to 
determine the attraction on each particle due to this displacement. 

The density at any point will differ from the original density by a quantity 
k' , so that 

{k + k') (dx + d^) = kdx (73), 

k'= —k-r- = Akm sin mx (74). 

The potential at any point will be V+V, where V is the original potential, 
and F' depends on the displacement only, so that 

dT d'V d'V ^ ,, ^ ,^,, 

^+-5^ + ^- + ^'^^=^ (^^)- 

Now V is a function of x only, and therefore, 

V = AirAk —sinmx (76), 

and the longitudinal force is found by differentiating V with respect to x. 

dV 
X= -,— = ink A cos mx = 'ink^ (77). 

Now let us suppose this mass not of infinite extent, but of finite section 
parallel to the plane of yz. This change amounts to cutting off all portions 
of the mass beyond a certain boundary. Now the effect of the portion so cut 
off upon the longitudinal force depends on the value of m. When m is large, 
so that the wave-length is small, the effect of the external portion is insensible, 
so that the longitudinal force due to short waves is not diminished by cutting 
off a great portion of the mass. 



338 ON THE STABILITY OF THE MOTION OF SATURN S RINGS. 

22. Applying this result to the case of a ring, and putting s for x, and 

a- for $ we have 

cr = ^ cos ms, and T= AttJcA cos ms, 

so that -RN=4:Trk, 

when on is very large, and this is the greatest value of N. 

The value of L has little effect on the condition of stability. If L and 
M are both neglected, that condition is 

(o'>27-S5e (2nk) (78), 

and if L be as much as ^N, then 

o>^>25-649 (27rk) (79), 

so that it is not important whether we calculate the value of L or not. 

The condition of stability is, that the average density must not exceed a 
certain value. Let us ascertain the relation between the maximum density of 
the ring and that of the planet. 

Let h be the radius of the planet, that of the ring being unity, then the 
mass of Saturn is ^Trh'k' = o)"' if k' be the density of the planet. If we assume 
that the radius of the ring is twice that of the planet, as Laplace has done, 
then h = ^ and 

1 = 334-2 to 307-7 (80), 

so that the density of the ring cannot exceed 3^ of that of the planet. Now 
Laplace has shewn that if the outer and inner parts of the ring have the same 
angular velocity, the ring will not hold together if the ratio of the density of 
the planet to that of the ring exceeds 1-3, so that in the first place, our ring 
cannot have uniform angular velocity, and in the second place, Laplace's ring 
cannot preserve its form, if it is composed of loose materials acting on each 
other only by the attraction of gravitation, and moving with the same angular 
velocity throughout. 

23. On the forces arising from inequalities of thickness in a thin stratum 
of fluid of indefinite extent. 

The forces which act on any portion of a continuous fluid are of two kinds, 
the pressures of contiguous portions of fluid, and the attractions of all portions of 
the fluid whether near or distant. In the case of a thin stratum of fluid, not 



ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 339 

acted on by any external forces, the pressures are due mainly to the component 
of the attraction which is perpendicular to the plane of the stratum. It is 
easy to shew that a fluid acted on by such a force will tend to assume a 
position of equilibrium, in which its free surface is plane ; and that any irregu- 
larities will tend to equalise themselves, so that the plane surface will be one 
of stable equilibrium. 

It is also evident, that if we consider only that part of the attraction 
which is parallel to the plane of the stratum, we shall find it always directed 
towards the thicker parts, so that the effect of this force is to draw the fluid 
from thinner to thicker parts, and so to increase irregularities and destroy 
equilibrium. 

The normal attraction therefore tends to preserve the stability of equilibrium, 
while the tangential attraction tends to render equilibrium unstable. 

According to the nature of the irregularities one or other of these forces 
will prevail, so that if the extent of the irregularities is small, the normal 
forces will ensure stability, while, if the inequaUties cover much space, the 
tangential forces will render equilibrium unstable, and break up the stratum into 
beads. 

To fix our ideas, let us conceive the irregularities of the stratum split up 
into the form of a number of systems of waves superposed on one another, 
then, by what we have just said, it appears, that very short waves will disap- 
pear of themselves, and be consistent with stability, while very long waves will 
tend to increase in height, and will destroy the form of the stratum. 

In order to determine the law according to which these opposite effects 
take place, we must subject the case to mathematical investigation. 

Let us suppose the fluid incompressible, and of the density k, and let it 
be originally contained between two parallel planes, at distances +c and — c 
from that of (xy), and extending to infinity. Let us next conceive a series of 
imaginary planes, parallel to the plane of {ijz), to be plunged into the fluid 
stratum at infinitesimal distances from one another, so as to divide the fluid 
into imaginary slices perpendicular to the plane of the stratum. 

Next let these planes be displaced parallel to the axis of x according to this 
law — that if x be the original distance of the plane from the origin, and ^ its 
displacement in the direction of x, 

i=A cosmx (81). 



340 ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 

According to this law of displacement, certain alterations will take place in 
the distances between consecutive planes ; but since the fluid is incompressible, 
and of indefinite extent in the direction of y, the change of dimension must 
occur in the direction of z. The original thickness of the stratum was 2c. Let 
its thickness at any point after displacement be 2c + 2^, then we must have 



.+i)=2^ («2)' 

1= — c -r-=cmA sinwa; (83). 



(2c + 20 (l 



Let us assume that the increase of thickness 2^ is due to an increase of C, 
at each surface ; this is necessary for the equilibrium of the fluid between the 
imaginary planes. 

We have now produced artificially, by means of these planes, a system of 

waves of longitudinal displacement whose length is — and amplitude A ; and 

we have found that this has produced a system of waves of normal displace- 
ment on each surface, having the same length, with a height =cmA. 

In order to determine the forces arising from these displacements, we must, 
in the first place, determine the potential function at any point of space, and 
this depends partly on the state of the fluid before displacement, and partly 
on the displacement itself We have, in all cases — 

d'V d'V d'V 
^^+^ + ^=-^^^ («^)- 

Within the fluid, p = k; beyond it, p = 0. 
Before displacement, the equation is reduced to 

d^' = -'-p («^)- 

Instead of assuming F=0 at infinity, we shall assume F=0 at the origin, 
and since in this case all is symmetrical, we have 

within the fluid F, = - 2nkz' -, ^ = - inJcz 



at the bounding planes F= — iirkc^ ; ->- = T 47r^c 

beyond them V, = 27r^c ( + 2z ± c) ; -y- = =F ^nkc 



.(86); 



ON THE STABILITY OF THE MOTION OF SATURN S RINGS. 341 

the upper sign being understood to refer to the boundary at distance +c, and 
the lower to the boundary at distance — c from the origin. 

Having ascertained the potential of the undisturbed stratum, we find that 
of the disturbance by calculating the effect of a stratum of density k and 
thickness t„ spread over each surface according to the law of thickness already 
found. By supposing the coeJB&cient A small enough, (as we may do in calcu- 
lating the displacements on which stabiUty depends), we may diminish the 
absolute thickness indefinitely, and reduce the case to that of a mere " super- 
ficial density," such as is treated of in the theory of electricity. We have here, 
too, to regard some parts as of negative density ; but we must recollect that we 
are dealing with the difference between a disturbed and an undisturbed system, 
which may be positive or negative, though no real mass can be negative. 

Let us for an instant conceive only one of these surfaces to exist, and let 
us transfer the origin to it. Then the law of thickness is 

l, = mcABm.'mx (83), 

and we know that the normal component of attraction at the surface is the 
same as if the thickness had been uniform throughout, so that 

on the positive side of the surface. 
Also, the solution of the equation 

d'V dyv_ 

dx" "^ dz' ~ ' 

consists of a series of terms of the form Ce'" sin ix. 

Of these the only one with which we have to do is that in which i= —m. 
Applying the condition as to the normal force at the surface, we get 

V=2'irkce''^Asmmx (87), 

for the potential on the positive side of the surface, and 

V=27rkce'^ABm7nx (88), 

on the negative side. 



[ (89) 



342 ON THE STABILITY OF THE MOTION OF SATURN S RINGS. 

Calculating the potentials of a pair of such surfaces at distances +c and —c 
from the plane of xy, and calling V the sum of their potentials, we have for 
the space between these planes 

F/ = 2TrkcA sin mxe""" (e"^ + e-*^) 
beyond them F/ = 27rZ;c^ sinma!;e^"^(e'^ + e~'^) 

the upper or lower sign of the index being taken according as z is positive or 
negative. 

These potentials must be added to those formerly obtained, to get the 
potential at any point after displacement. 

We have next to calculate the pressure of the fluid at any point, on the 
supposition that the imaginary planes protect each shce of the fluid from the 
pressure of the adjacent sHces, so that it is in equilibrium under the action of 
the forces of attraction, and the pressure of these planes on each side. Now 
in a fluid of density h, in equilibrium under forces whose potential is V, we 
have always — 

so that if we know that the value of p is 2\ where that of F is F^, then at 

any other point 

jD=^„ + ^(F-F„). 

Now, at the free surface of the fluid, ]p = 0, and the distance from the 
free surface of the disturbed fluid to the plane of the original surface is ^, a 
small quantity. The attraction which acts on this stratum of fluid is, in the 
first place, that of the undisturbed stratum, and this is equal to A^irkc, towards 
that stratum. The pressure due to this cause at the level of the original 
surface will be AnJifcC, and the pressure arising from the attractive forces due 
to the displacements upon this thin layer of fluid, will be small quantities of 
the second order, which we neglect. We thus find the pressure when z = c to be, 

Pa = AvJc^c^mA sin mx. 

The potential of the undisturbed mass when z = c is 

V,= -2TTkc\ 
and the potential of the disturbance itself for the same value of z, is 

F; = 2TrkcA sin mx (1 + e""^). 



ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 343 

So that we find the general value of jp at any other point to be 

^ = 27r^^ (c' - z') + 27r/:'c^ sin ?7ia; {2c»i - 1 - €- ^"^ + e"^ (e"- + e""^)} . . . (90). 

This expression gives the pressure of the fluid at any point, as depending 
on the state of constraint produced by the displacement of the imaginary planes. 
The accelerating effect of these pressures on any particle, if it were allowed to 
move parallel to x, instead of being confined by the planes, would be 

_1 dp 
k dx' 

The accelerating effect of the attractions in the same direction is 

dV 
dx' 

so that the whole acceleration parallel to cc is 

X= -lirkmcA cos 7nx {2mc - e''^ - I) (91). 

It is to be observed, that this quantity is independent of z, so that every 
particle in the slice, by the combined effect of pressure and attraction, is urged 
with the same force, and, if the imaginary planes were removed, each slice 
would move parallel to itself without distortion, as long as the absolute dis- 
placements remained small. We have now to consider the direction of the 
resultant force X, and its changes of magnitude. 

We must remember that the original displacement is A cos 7nx, if therefore 
(2mo-e~"^— 1) be positive, X will be opposed to the displacement, and the 
equilibrium will be stable, whereas if that quantity be negative, X will act 
along with the displacement and increase it, and so constitute an unstable 
condition. 

It may be seen that large values of nic give positive results and small 
ones negative. The sign changes when 

2mc = l'lA7 (92), 

which corresponds to a wave-length 

\ = 2c^^^ = 2c{5'i7l) (93). 

The length of the complete wave in the critical case is 5*471 times the 
thickness of the stratum. Waves shorter than this are stable, longer waves 
are unstable. 



344 ON THE STABILITY OF THE MOTION OF SATURN S RINGS. 

The quantity 2mc{2mc-e-^-l), 

has a minimum when 2mc = '607 (94), 

and the wave-length is 10 '3 5 3 times the thickness of the stratum. 

In this case 2mc (2mc-e-^"^- 1)= - '509 (95), 

and X='5097rMcosmx (96). 

24. Let us now conceive that the stratum of fluid, instead of being infinite 
in extent, is limited in breadth to about 100 times the thickness. The pressures 
and attractions will not be much altered by this removal of a distant part of 
the stratum. Let us also suppose that this thin but broad strip is bent round 
in its own plane into a circular ring whose radius is more than ten times the 
breadth of the strip, and that the waves, instead of being exactly parallel to 
each other, have their ridges in the direction of radii of the ring. We shall 
then have transformed our stratum into one of Saturn's Kings, if we suppose 
those rings to be liquid, and that a considerable breadth of the ring has the 
same angular velocity. 

Let us now investigate the conditions of stability by putting 
x= - 27rkmc (2mc - e"^ - 1) 

into the equation for n. We know that x must lie between and ^^ to 

ensure stabihty. Now the greatest value of x in the fluid stratum is -50917^-. 
Taking Laplace's ratio of the diameter of the ring to that of the planet, this 
gives 42-5 as the minimum value of the density of the planet divided by that 
of the fluid of the ring. 

Now Laplace has shewn that any value of this ratio greater than 1-3 is 
inconsistent with the rotation of any considerable breadth of the fluid at the 
same angular velocity, so that our hypothesis of a broad ring with uniform 
velocity is untenable. 

But the stabihty of such a ring is impossible for another reason, namely, 
that for waves in which 2mc> 1-147, x is negative, and the ring will be destroyed 
by these short waves in the manner described at page (333). 

When the fluid ring is treated, not as a broad strip, but as a filament of 
circular or elliptic section, the mathematical difiSculties are very much increased. 



ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 345 

but it may be shown that in this case also there will be a maximum value 
of X, which will require the density of the planet to be several times that of 
the ring, and that in all cases short waves will give rise to negative values 
of X, inconsistent with the stability of the rmg. 

It appears, therefore, that a ring composed of a continuous liquid mass 
cannot revolve about a central body without being broken up, but that the 
parts of such a broken ring may, under certain conditions, form a permanent 
ring of satellites. 

On the Mutual Perturbations of Two Rings. 

25. We shall assume that the difference of the mean radii of the rings 
is small compared with the radii themselves, but large compared with the 
distance of consecutive satellites of the same ring. We shall also assume that 
each ring separately satisfies the conditions of stability. 

We have seen that the effect of a disturbing force on a ring is to produce 
a series of waves whose number and period correspond with those of the dis- 
turbing force which produces them, so that we have only to calculate the 
coefficient belonging to the wave from that of the disturbing force. 

Hence in investigating the simultaneous motions of two rings, we may 
assume that the mutually disturbing waves travel with the same absolute 
angular velocity, and that a maximum in one corresponds either to a maximum 
or a minimum of the other, according as the coefficients have the same or 
opposite signs. 

Since the motions of the particles of each ring are affected by the disturbance 
of the other ring, as well as of that to which they belong, the equations of 
motion of the two rings will be involved in each other, and the final equation 
for determining the wave-velocity will have eight roots instead of four. But as 
each of the rings has four free waves, we may suppose these to originate forced 
waves in the other ring, so that we may consider the eight waves of each ring 
as consisting of four free waves and four forced ones. 

In strictness, however, the wave- velocity of the "free" waves will be 
affected by the existence of the forced waves which they produce in the other 
ring, so that none of the waves are really " free " in either ring independently, 
though the whole motion of the system of two rings as a whole is free. 



346 ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 

We shall find, however, that it is best to consider the waves first as free, 
and then to determine the reaction of the other ring upon them, which is such 
as to alter the wave-velocity of both, as we shall see. 

The forces due to the second ring may be separated into three parts. 

1st. The constant attraction when both rings are at rest. 

2nd. The variation of the attraction on the first ring, due to its own 
disturbances. 

3rd. The variation of the attraction due to the disturbances of the second 
ring. 

The first of these affects only the angular velocity. The second affects the 
waves of each ring independently, and the mutual action of the waves depends 
entirely on the third class of forces. 

26. To deteivnine the attractions between two rings. 

Let R and a be the mass and radius of the exterior ring, R and a' those 
of the interior, and let all quantities belonging to the interior ring be marked 
with accented letters. (Fig. 5.) 

1st. Attraction between the rings when at rest. 

Since the rings are at a distance small compared with their radii, we may 
calculate the attraction on a particle of the first ring as if the second were an 
infinite straight line at distance a' — a from the first. 

7?' 

The mass of unit of length of the second ring is - — > , and the accelerating 

effect of the attraction of such a filament on an element of the first ring is 

TV 

— —, 7\ inwards (97). 

na [a — a) ^ 

The attraction of the first ring on the second may be found by transposing 
accented and unaccented letters. 

In consequence of these forces, the outer ring will revolve faster, and the 

inner ring slower than would otherwise be the case. These forces enter into 

the constant terms of the equations of motion, and may be included in the 
value of K. 



ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 347 

2nd. Variation due to disturbance of first ring. 

If we put a(l+p) for a in the last expression, we get the attraction 
when the first ring is displaced. The part depending on p is 

r-, TT, P inwards (98). 

Tra [a-ay '^ 

This is the only variation of force arising from the displacement of the 
first ring. It affects the value of X in the equations of motion. 

3rd. Variation due to waves in the second ling. 

On account of the waves, the second ring varies in distance from the 
first, and also in mass of unit of length, and each of these alterations produces 
variations both in the radial and tangential force, so that there are four things 
to be calculated : 

1st. Radial force due to radial displacement. 

2nd, Radial force due to tangential displacement. 

3rd. Tangential force due to radial displacement. 

4th. Tangential force due to tangential displacement. 

1st. Put a'(l+p') for a\ and we get the term in p 

— -, \ ? ~ ,; p' inwards = XV> say (99). 

ira (a -af ^ t^ > J v ^ 



2nd. By the tangential displacement of the second ring the section is 
iced in the proportion 
of the radial force equal to 



reduced in the proportion of 1 to l--j , , and therefore there is an alteration 



-yr inwards = — /x' -j-, say (100). 



ird'(a — a') ds' '^ ds' 

3rd. By the radial displacement of the second ring the direction of the 
filament near the part in question is altered, so that the attraction is no longer 
radial but forwards, and the tangential part of the force is 

.5 '^ ^'=+/^' forwards (lOl). 

ira (a-a) ds '^ ds 

44—2 



348 ON THE STABILITY OF THE MOTION OF SATURN S RINGS. 

4th. By the tangential displacement of the second ring a tangential force 

arises, depending on the relation between the length of the waves and the 

distance between the rings. 

"-ot' J f+«xsinp^ , 

If we make m — - = p, and m -i ax = H, 

a ^ J-o.(l+x-y 

/?' 

the tangential force is a (a-a'Y ^^' ^ *''^' (102). 

We may now write down the values of X, /x, and v by transposing accented 
and unaccented letters. 

g^(2a-a) R ^^ _?_ n (103). 

ira (a-aj '^ TTa{a-a)' ira {a-af 

Comparing these values with those of X', /x', and v, it will be seen that 
the following relations are approximately true when a is nearly equal to a: 

^'=-'i = ^ = |> (104). 

X H' ^ R^ 

27. To form the equations of motion. 

*The original equations were 

^■' + o,'p + -2o,^-'^, = P = S+K-(2S-L)Ap-MBp + yp--y:'^, 

Putting p = ^ cos {ills + nt), ar = B8m (ms + nt), 

p' = A' cos {im + nt), cr'^R sin {ins + nt), 
then u>' = S-vK 

{(o'■V2S+n'-L)A + {2(on+M)B-XA' + |J:mB = 0^ ,^^^. 

{2con + M)A + {n'^-N)B-ij:mA' + vR = o] ^ '' 

The corresponding equations for the second ring may be found by trans- 
posing accented and unaccented letters. We should then have four equations 
to determine the ratios of A, B, A', B', and a resultant equation of the eighth 
degree to determine n. But we may make use of a more convenient method, 
since X', ix, and v are small. Eliminating B we find 

An'-A(ai'^-lK+L-N)n'-iAo>Mn + AN{Zoy)\_ , . 

(-X'A' + fx'mR)n' + {ix'mA' -v'B') 2<onj ^ '' 

* [The analysis in this article is somewhat unsatisfactory, the equations of motion employed being 
those which were applicable in the case of a ring of radius unity. Ed.] 



ON THE STABILITY OP THE MOTION OF SATURN's RINGS. 349 

Putting B = ^A, A' = xA, B' = ^A' = ^xA, 

we have ii* - {o.' ( + 2 A") + X - iV} n' - 4(oMn + Sco'N] ^jj^^ / ^qj^x 

~ = 47i'-2a;';i + &c (108), 

an 

-r = - ^''^' + H''ml3'}r + 2/»iw?i - 2u^'a)n (109), 

28. If we were to solve the equation for n, leaving out the terms involving 
X, we should find the wave-velocities of the four free waves of the first ring, 
supposing the second ring to be prevented from being disturbed. But in reality 
the waves in the first ring produce a disturbance in the second, and these in 
turn react upon the first ring, so that the wave-velocity is somewhat difierent 
from that which it would be in the supposed case. Now if x be the ratio 
of the radial amplitude of displacement in the second ring to that in the first, 
and if n be a value of n supposing cc = 0, then by Maclaurin's theorem, 

n= Jfn + -j-x (Ill)- 

The wave-velocity relative to the ring is , and the absolute angular 

velocity of the wave in space is 

n n I dn . ^-. 

'ar = oi =0) j-x (112), 

m m m ax ' 

= +p-qx (113), 

, n , \ dn 

where » = w , and o = — -j- . 

^ m ^ m ax 

Similarly in the second ring we should have 

-=/-<z'^ (114); 

and since the corresponding waves in the two rings must have the same abso- 
lute angular velocity, 

^ = ■25-', or 'p — qx^'p—ci - (115)- 



350 ON THE STABILITY OF THE MOTION OF SATURN S RINGS, 

This is a quadratic equation in x, the roots of which are real when 

is positive. When this condition is not fulfilled, the roots are impossible, and 
the general solution of the equations of motion will contain exponential factors, 
indicating destructive oscillations in the rings. 

Since q and q' are small quantities, the solution is always real whenever 
p and p' are considerably different. The absolute angular velocities of the two 
pairs of reacting waves, are then nearly 

V -\ — ^^/ , and r) — ^^, , 

instead of p and p\ as they would have been if there had been no reaction 
of the forced wave upon the free wave which produces it. 

When 2^ and p' are equal or nearly equal, the character of the solution 
will depend on the sign of qq. We must therefore determine the signs of q 
and q' in such cases. 

Putting P = —7-, we may write the values of q and q' 






x/ ^ / /6> fO\ ,,(0 0) 

X + 211 m — - - - 4i/ - - 
n ^ \n 71/ 71 71 



7n ' 4?i^ — 2<xr 



Oi Ct/\ , Oi 0) 

, _ n ^ \n 71 1 71 n 

^~m" in"-2o)" 

Referring to the values of the disturbing forces, we find that 

X' IX V _ Ka 
X iL V Ra" 



(116). 



TT g n 471* — 2&> Ra l^^*7\ 

Hence X = _^ — , —-, (117). 

q n 4n'-2w* Ra 

Since qq' is of the same sign as -^ , we have only to determine whether 

2 '2 

2n--, and 2n' -— , are of the same or of different signs. If these quantities 
n 71 ' 

are of the same sign, qq is positive, if of different signs, qq' is negative. 



ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 351 

Now there are four values of n, which give four corresponding values of 



2n 



72 1= -W + &C., 2?ii- is negative, 
??j = — a small quantity, 2n^ is positive, 



jjj = -f. a small quantity, 211^ is negative, 

^3 



n^ = oi — kc., 271^ is positive. 



The quantity with which we have to do is therefore positive for the even 
orders of waves and negative for the odd ones, and the corresponding quantity 
in the other ring obeys the same law. Hence when the waves which act upon 
each other are either both of even or both of odd names, qq will be positive, 
but when one belongs to an even series, and the other to an odd series, qq 
is negative. 



29. The values of j) and p' are, roughly, 



X>^ = oi + — — &c., ^o = w + &c., ^3 = (u — &c., ^4 = (o — — - + &c. 

^j' = Co' H &C., p.' = 0) + &c., Pa' = co' — &c., Pi=Oi 1- &C. 



(118). 



<ji is greater than <u, so that j>^ is the greatest, and Pi the least of these 
values, and of those of the same order, the accented is greater than the unac- 
cented. The following cases of equahty are therefore possible under suitable 
circumstances ; 

P, =P,\ Pi =p/» 

P4=P,' (when m=l), p,=2^3, 

p.=p:, 

In the cases in the first column qq' will be positive, in those in the second 
column qq' will be negative. 



352 ox THE STABILITY OF THE MOTION OF SATURN's RINGS. 

30. Now each of the four values of p is a function of w, the number 
of undulations in the ring, and of a the radius of the ring, varying nearly 
as cfl Hence m being given, we may alter the radius of the ring till any 
one of the four values of p becomes equal to a given quantity, say a given 
value of /, so that if an indefinite number of rings coexisted, so as to form 
a sheet of rings, it would be always possible to discover instances of the 
equality of x> ^^^ V among them. K such a case of equahty belongs to the 
first column given above, two constant waves will arise in both rings, one 
travelling a little faster, and the other a little slower than the free waves. 
If the case belongs to the second column, two waves will also arise in each 
ring, but the one pair will graduaUy die away, and the other pair wHl increase 
in ampUtude indefinitely, the one wave strengthening the other till at last both 
rino-s are thrown into confusion. 

The only way in which such an occurrence can be avoided is by placing 
the rings at such a distance that no value of m shall give coincident values 
of _p and J), For instance, if w > 2a), but w < So), no such coincidence is possible. 
For j)^ is always less than p./, it is greater than p, when m = 1 or 2, and less 
than _p4 when m is 3 or a greater number. There are of course an infinite 
number of ways in which this noncoincidence might be secured, but it is plain 
that if a number of concentric rings were placed at small intervals from each 
other, such coincidences must occur accurately or approximately between some 
pairs' of rings, and if the value of [p-fj is brought lower than -^qq, there 
will be destructive interference. 

This investigation is applicable to any number of concentric rings, for, by 
the principle of superposition of small displacements, the reciprocal actions of 
any pair of rings are independent of all the rest. 

31. On the effect of long-continued disturbances on a system of rings. 

The result of our previous investigations has been to point out several 
ways in which disturbances may accumulate till collisions of the different par- 
ticles of the rings take place. After such a collision the particles wUl still 
continue to revolve about the planet, but there will be a loss of energy in 
the system during the colUsion which can never be restored. Such coUisions 
however will not affect what is called the Angular Momentum of the system 
about the planet, which will therefore remain constant. 



ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 353 

Let M be the mass of tlie system of rings, and hm that of one ring 
whose radius is r, and angular velocity (o = S^r~^. The angular momentum of 
the ring is 

half its vis viva is ^tuV'Sm = ^Sr~^ hm. 

The potential energy due to Saturn's attraction on the ring is 

-Sr-'hm. 
The angular momentum of the whole system is invariable, and is 

S'^%{r^hm) = A (119). 

The whole energy of the system is the sum of half the vis viva and the 
potential energy, and is 

-^St{r-'hm) = E (120). 

A is invariable, while E necessarily diminishes. We shall find that as E 
diminishes, the distribution of the rings must be altered, some of the outer 
rings moving outwards, while the inner rings move inwards, so as either to 
spread out the whole system more, both on the outer and on the inner edge 
of the system, or, without affecting the extreme rings, to diminish the density 
or number of the rings at the mean distance, and increase it at or near the 
inner and outer edges. 



Let us put x = r^-, 


then A- 


= S-t{xdm) is constant. 


Now let 




tixdm) 
^^~ t{dm) ' 


and 




X = Xi + x\ 


then we may write 







-^ = t(r-^Bm)=^t{x-'dm), 
lb 



= Sc^m(a:--2|3 + 3|i-&c.), 



= \t{dm)-^,X{xdm)-]-^,t(x'Bm)-kc (121). 



354 ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 

Now t(dm) = M a constant, t(xdm) = 0, and t(x"-Bm) is a quantity which 
increases when the rings are spread out from the mean distance either way, 
X being subject only to the restriction t (xdm) = 0. But % (x'dm) may 
increase without the extreme values of x being increased, provided some other 
values be increased. 

32. In fact, if we consider the very innermost particle as moving in an 
ellipse, and at the further apse of its orbit encountering another particle 
belonging to a larger orbit, we know that the second particle, when at the 
same distance from the planet, moves the faster. The result is, that the 
interior satellite will receive a forward impulse at its further apse, and will 
move in a larger and less eccentric orbit than before. In the same way one 
of the outermost particles may receive a backward impulse at its nearer apse, 
and so be made to move in a smaller and less eccentric orbit than before. 
When we come to deal with collisions among bodies of unknown number, size, 
and shape, we can no longer trace the mathematical laws of their motion with 
any distinctness. All we can now do is to collect the results of our investi- 
gations and to make the best use we can of them in forming an opinion as 
to the constitution of the actual rings of Saturn which are still in existence 
and apparently in steady motion, whatever catastrophes may be indicated by 
the various theories we have attempted. 

33. To find the Loss of Energy due to internal friction in a hroad Fluid 
Ring, the parts of which revolve about the Planet, each with the velocity of a 
satellite at the same distance. 

Conceive a fluid, the particles of which move parallel to the axis of x 
with a velocity u, u being a function of z, then there will be a tangential pres- 
sure on a plane parallel to xy 

dU .. r. 

= /x-y- on umt 01 area 
'^ dz 

due to the relative sliding of the parts of the fluid over each other. 
In the case of the ring we have 

The absolute velocity of any particle is tor. That of a particle at distance 
{r-\-Zr) is 

(ar + -j- {(ar) hr. 



ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 355 

If the angular velocity had been uniform, there would have been no sliding, 
and the velocity would have been 

cji" + (ohr. 
The sliding is therefore 

d(o ^ 

r -J- or, 
ar 

and the friction on unit of area perpendicular to r is fir -p • 

The loss of Energy, per unit of area, is the product of the sliding by the 
friction, 

or, /x?-*-^ Sr in unit of time. 

The loss of Energy in a part of the Ring whose radius is r, breadth 
Sr, and thickness c, is 

27rr*c/x -j- Sr. 
In the case before us it is f Tr/x/Scr"* Sr. 

If the thickness of the ring is uniform between r = a and r = h, the whole 
loss of Energy is 

in unit of time. 

Now half the vis viva of an elementary ring is 
npcrhr r^oy = nfxSSr, 
and this between the limits r = a and r = h gives 

npcS (a — h). 

The potential due to the attraction of 5 is twice this quantity with the 
sign changed, so that 

E=-TrpcS(a-b), 

E dt~ ^ p ah' 

45—2 



356 ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 

Now Professor Stokes finds a/^ = 0-0564 for water, 

^ P 

and =0'116 for air, 

taking the unit of space one English inch, and the unit of time one second. 
We may take a = 88,209 miles, and ?> = 77,636 for the ring A) and a = 75,845, 
and 6 = 58,660 for the ring B. We may also take one year as the unit of 
time. The quantity representing the ratio of the loss of energy in a year to 
the whole energy is 

I dE 1 p .-L • ^ 

E W= 60,880,000,000,000 ^^' ^^' "^^ ^' 

^^ 39,540,000,000,000 ^'' ^^^ ^^"^ ^' 

showing that the efiect of internal friction in a ring of water moving with 
steady motion is inappreciably small. It cannot be from this cause therefore 
that any decay can take place in the motion of the ring, provided that no 
waves arise to disturb the motion. 



Recapitulation of the Tlieory of the Motion of a Rigid Ring. 

The position of the ring relative to Saturn at any given instant is defined 
by three variable quantities. 

1st. The distance between the centre of gravity of Saturn and the centre 
of gravity of the ring. This distance we denote by r. 

2nd. The angle which the line r makes with a fixed line in the plane of 
the motion of the ring. This angle is called 0. 

3rd. The angle between the line r and a Hne fixed with respect to the 
ring so that it coincides with r when the ring is in its mean position. This is 
the angle <^. 

The values of these three quantities determine the position of the ring so 
far as its motion in its own plane is concerned. They may be referred to as 
the radius vector, longitude, and angle of lihration of the ring. 

The forces which act between the ring and the planet depend entirely upon 
their relative positions. The method adopted above consists in determining the 



ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 357 

potential ( V) of the ring at the centre of the planet In terms of r and <^. Then 

the work done by any displacement of the system is measured by the change 

of VS during that displacement. The attraction between the centre of gravity 

(IV 
of the Ring and that of the planet Is ~S , , and the moment of the couple 

clV 
tending to turn the ring about Its centre of gravity Is S-j-j, 

It Is proved In Problem V, that if a be the radius of a circular ring, r^^uf 
the distance of its centre of gravity from the centre of the circle, and R the 

mass of the ring, then, at the centre of the ring, ,- = 5/, -yj = 0. 

(PV Ji 
It also appears that T-^ = -k~3 {^ +9)> "which is positive when g > —I, 

d'V R 
and that -n\=^—f'(^—g), which is positive when ^<3. 

d'V . . . 

If -y— is positive, then the attraction between the centres decreases as the 

distance increases, so that, if the two centres were kept at rest at a given 

d'V . . . 
distance by a constant force, the equilibrium would be unstable. If -t-t; is positive, 

then the forces tend to increase the angle of libration, in whichever direction 
the libration takes place, so that if the ring were fixed by an axis through its 
centre of gravity, its equilibrium round that axis would be unstable. 

In the case of the uniform ring with a heavy particle on its circumference 
whose weight ="82 of the whole, the direction of the whole attractive force of 
the ring near the centre will pass through a point lying in the same radius as 
the centre of gravity, but at a distance from the centre = fa. (Fig. 6.) 

If we call this point 0, the line SO will indicate the direction and position 
of the force acting on the ring, which we may call F. 

It Is evident that the force F, acting on the ring in the line OS, will tend 
to turn it round its centre of gravity R and to increase the angle of libration 
KRO. The direct action of this force can never reduce the angle of libration 
to zero again. To understand the indirect action of the force, we must recollect 
that the centre of gravity (i?) of the ring is revolving about Saturn in the 
direction of the arrows, and that the ring is revolving about its centre of gravity 



358 ON THE STABILITY OF THE MOTION OF SATURN S RINGS. 

with nearly the same velocity. If the angular velocity of the centre of gravity 
about Saturn were always equal to the rotatory velocity of the ring, there 
would be no libration. 

Now suppose that the angle of rotation of the ring is in advance of the 
longitude of its centre of gravity, so that the line RO has got in advance of 
SRK by the angle of libration KRO. The attraction between the planet and 
the ring is a force F acting in SO. We resolve this force into a couple, whose 
moment is FRN, and a force F acting through R the centre of gravity of the 
ring. 

The couple affects the rotation of the ring, but not the position of its centre 
of gravity, and the force RF acts on the centre of gravity without affecting the 
rotation. 

Now the couple, in the case represented in the figure, acts in the positive 
direction, so as to increase the angular velocity of the ring, which was already 
greater than the velocity of revolution of R about S, so that the angle of 
libration would increase, and never be reduced to zero. 

The force RF does not act in the direction of >S', but behind it, so that it 
becomes a retarding force acting upon the centre of gravity of the ring. Now 
the effect of a retarding force is to cause the distance of the revolving body to 
decrease and the angular velocity to increase, so that a retarding force increases 
the angular velocity of R about S. 

The effect of the attraction along SO in the case of the figure is, first, to 
increase the rate of rotation of the ring round R, and secondly, to iacrease the 
angular velocity of R about S. If the second effect is greater than the first, 
then, although the line RO increases its angular velocity, SR will increase its 
angular velocity more, and will overtake RO, and restore the ring to its original 
position, so that SRO will be made a straight line as at first. If this accelerat- 
ing effect is not greater than the acceleration of rotation about R due to the 
couple, then no compensation will take place, and the motion will be essentially 
unstable. 

If in the figure we had drawn ^ negative instead of positive, then the 
couple would have been negative, the tangential force on R accelerative, r would 
have increased, and in the cases of stability the retardation of 6 would be greater 
than that of (^ + <^), and the normal position would be restored, as before. 



ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 359 

The object of the investigation is to find the conditions under wliich this 
compensation is possible. 

It is evident that when SRO becomes straight, there is still a difference 
of angular velocities between the rotation of the ring and the revolution of 
the centre of gravity, so that there will be an oscillation on the other side, 
and the motion will proceed by alternate oscillations without limit. 

If we begin with r at its mean value, and <^ negative, then the rotation 
of the ring will be retarded, 7* will be increased, the revolution of r will be 
more retarded, and thus <f> will be reduced to zero. The next part of the 
motion will reduce r to its mean value, and bring (f) to its greatest positive 
value. Then r will diminish to its least value, and (f> will vanish. Lastly r 
will return to the mean value, and <f) to the greatest negative value. 

It appears from the calculations, that there are, in general, two different 
ways in which this kind of motion may take place, and that these may have 
different periods, phases, and amplitudes. The mental exertion required in follow- 
ing out the results of a combined motion of this kind, with all the variations of 
force and velocity during a complete cycle, w^ould be very great in proportion to 
the additional knowledge we should derive from the exercise. 

The result of this theory of a rigid ring shows not only that a perfectly 
uniform ring cannot revolve permanently about the planet, but that the irregu- 
larity of a permanently revolving ring must be a very observable quantity, the 
distance between the centre of the ring and the centre of gravity being between 
•8158 and '8279 of the radius. As there is no appearance about the rings 
justifying a belief in so great an irregularity, the theory of the solidity of the 
rings becomes very improbable. 

When we come to consider the additional difficulty of the tendency of the 
fluid or loose parts of the ring to accumulate at the thicker parts, and thus 
to destroy that nice adjustment of the load on which stability depends, we 
have another powerful argument against solidity. 

And when we consider the immense size of the rings, and their comparative 
thinness, the absurdity of treating them as rigid bodies becomes self-evident. 
An iron ring of such a size would be not only plastic but semifluid under the 
forces which it would experience, and we have no reason to believe these rings 
to be artificially strengthened with any material unknown on this earth. 



360 ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 



Recapitulation of the Theory of a Ring of equal Satellites. 

In attempting to conceive of the disturbed motion of a ring of unconnected 
satellites, we have, in the first place, to devise a method of identifying each 
satellite at any given time, and in the second place, to express the motion of 
every satellite under the same general formula, in order that the mathematical 
methods may embrace the whole system of bodies at once. 

By conceiving the ring of satellites arranged regularly in a circle, we may 
easily identify any satellite, by stating the angular distance between it and a 
known satellite when so arranged. If the motion of the ring were undisturbed, 
this angle would remain unchanged during the motion, but, in reality, the 
satellite has its position altered in three ways : 1st, it may be further from 
or nearer to Saturn; 2ndly, it may be in advance or in the rear of the position 
it would have had if undisturbed ; 3rdly, it may be on one side or other of 
the mean plane of the ring. Each of these displacements may vary in any way 
whatever as we pass from one satellite to another, so that it is impossible 
to assign beforehand the place of any satellite by knowing the places of the 
rest. § 2. 

The formula, therefore, by which we are enabled to predict the place of 
every satellite at any given time, must be such as to allow the initial position 
of every satellite to be independent of the rest, and must express all future 
positions of that satellite by inserting the corresponding value of the quantity 
denoting time, and those of every other sateUite by inserting the value of the 
angular distance of the given satelUte from the point of reference. The three 
displacements of the satellite will therefore be functions of two variables — the 
angular position of the satellite, and the time. When the time alone is made 
to vary, we trace the complete motion of a single satellite ; and when the time 
is made constant, and the angle is made to vary, we trace the form of the 
ring at a given time. 

It is evident that the fonn of this function, in so far as it indicates the 
state of the whole ring at a given instant, must be wholly arbitrary, for the 
form of the ring and its motion at starting are limited only by the condition 
that the irregularities must be small. We have, however, the means of breaking 
up any function, however complicated, into a series of simple functions, so that 
the value of the function between certain limits may be accurately expressed 



ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 361 

as the sum of a series of sines and cosines of multiples of the variable. This 
method, due to Fourier, is peculiarly applicable to the case of a ring returning 
into itself, for the value of Fourier's series is necessarily periodic. We now 
regard the form of the disturbed ring at any instant as the result of the 
superposition of a number of separate disturbances, each of -which is of the nature 
of a series of equal waves regularly arranged round the. ring. Each of these 
elementary disturbances is characterised by the number of undulations in it, by 
their amplitude, and by the position of the first maximum in the ring. § 3. 

When we know the form of each elementary disturbance, we may calculate 
the attraction of the disturbed ring on any given particle in terms of the con- 
stants belonging to that disturbance, so that as the actual displacement is the 
resultant of the elementary displacements, the actual attraction will be the 
resultant of the corresponding elementary attractions, and therefore the actual 
motion will be the resultant of all the motions arising from the elementary 
disturbances. We have therefore only to investigate the elementary disturbances 
one by one, and having established the theory of these, we calculate the actual 
motion by combining the series of motions so obtained. 

Assuming the motion of the satellites in one of the elementary disturbances 
to be that of oscillation about a mean position, and the whole motion to be 
that of a uniformly revolving series of undulations, we find our supposition to 
be correct, provided a certain biquadratic equation is satisfied by the quantity 
denoting the rate of oscillation. § 6. 

When the four roots of this equation are all real, the motion of each 
satellite is compounded of four difierent oscillations of difi'erent amplitudes and 
periods, and the motion of the whole ring consists of four series of undulations, 
travelling round the ring with different velocities. When any of these roots 
are impossible, the motion is no longer oscillatory, but tends to the rapid 
destruction of the ring. 

To determine whether the motion of the ring is permanent, we must assure 
ourselves that the four roots of this equation are real, whatever be the number 
of undulations in the ring; for if any one of the possible elementary distui'b- 
ances should lead to destructive oscillations, that disturbance might sooner or 
later commence, and the ring would be destroyed. 

Now the number of undulations in the ring may be any whole number 
from one up to half the number of satellites. The forces from which danger 



362 ON THE STABILITY OF THE MOTION OF SATURN S RINGS. 

is to be apprehended are greatest when the number of undulations is greatest, 
and by taking that number equal to half the number of satellites, we find the 
condition of stability to be 

S>.A352tiR, 

where S is the mass of the central body, R that of the ring, and /x the number 
of sateUites of which it is composed. § 8. If the number of satelHtes be too 
great, destructive oscillations will commence, and finally some of the satellites 
will come into coUision with each other and unite, so that the number of 
independent satellites will be reduced to that which the central body can retain 
and keep in discipline. When this has taken place, the satellites will not only 
be kept at the proper distance from the primary, but will be prevented by its 
preponderating mass from interfering with each other. 

We next considered more carefully the case in which the mass of the ring 
is very small, so that the forces arising from the attraction of the ring are 
small compared with that due to the central body. In this case the values 
of the roots of the biquadratic are all real, and easUy estimated. § 9. 

If we consider the motion of any satellite about its mean position, as 
referred to axes fixed in the plane of the ring, we shall find that it describes 
an ellipse in the direction opposite to that of the revolution of the ring, the 
periodic time being to that of the ring as o> to n, and the tangential ampli- 
tude of oscillation being to the radial as 2(0 to n. § 10. 

The absolute motion of each satellite in space is nearly elliptic for the large 
values of n, the axis of the ellipse always advancing slowly in the direction of 
rotation. The path of a satellite corresponding to one of the small values of 
n is nearly circular, but the radius slowly increases and diminishes during a 
period of many revolutions. § 11. 

The form of the ring at any instant is that of a re-entering curve, having 
m alternations of distance from the centre, symmetrically arranged, and m points 
of condensation, or crowding of the satellites, which coincide with the points of 
greatest distance when n is positive, and with the points nearest the centre 
when n m negative. § 12. 

This system of undulations travels with an angular velocity relative to 

the ring, and co in space, so that during each oscillation of a satellite a 

complete wave passes over it. § 14. 



ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 363 

To exhibit the movements of the satellites, I have made an arrangement 
by which 36 little ivory balls are made to go through the motions belonging 
to the first or fourth series of waves. (Figs. 7, 8.) 

The instrument stands on a pillar A, in the upper part of which turns 
the cranked axle CC. On the parallel parts of this axle are placed two wheels, 
RR and TT, each of which has 36 holes at equal distances in a circle neai- 
its circumference. The two circles are connected by 36 small cranks of the 
fonn KK, the extremities of which turn in the corresponding holes of the two 
wheels. That axle of the crank K which passes through the hole in the wheel 
S is bored, so as to hold the end of the bent wire which carries the satellite >S'. 
This wire may be turned in the hole so as to place the bent part carrying 
the satellite at any angle with the crank. A pin F, which passes through the 
top of the pillar, serves to prevent the cranked axle from turning ; and a pin Q, 
passing through the pillar horizontally, may be made to fix the wheel R, by 
inserting it in a hole in one of the spokes of that wheel. There is also a 
handle H, which is in one piece with the wheel T, and serves to turn the axle. 

Now suppose the pin P taken out, so as to allow the cranked axle to 
turn, and the pin Q inserted in its hole, so as to prevent the wheel R from 
revolving; then if the crank C be turned by means of the handle H, the 
wheel T will have its centre carried round in a vertical circle, but will remain 
parallel to itself during the whole motion, so that every point in its plane will 
describe an equal circle, and all the cranks K will be made to revolve exactly 
as the large crank C does. Each satellite will therefore revolve in a small 
circular orbit, in the same time with the handle H, but the position of each 
satellite in that orbit may be arranged as we please, according as we turn the 
wire which supports it in the end of the crank. 

In fig. 8, which gives a front view of the instrument, the satelHtes are so 
placed that each is turned 60^ further round in its socket than the one behind 
it. As there are 36 satellites, this process will bring us back to our starting- 
point after six revolutions of the direction of the arm of the satellite; and 
therefore as we have gone round the ring once in the same direction, the ami 
of the sateUite will have overtaken the radius of the ring five times. 

Hence there will be five places where the satellites are beyond their mean 
distance from the centre of the ring, and five where they are within it, so 
that we have here a series of five undulations round the circumference of the 

46—2 



364 ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 

ring. In this case the satellites are crowded together when nearest to the centre, 
so that the case is that of the first series of waves, when m = 5. 

Now suppose the cranked axle C to be turned, and all the small cranks 
K to turn with it, as before explained, every satellite will then be carried 
round on its own arm in the same direction ; but, since the direction of the 
arms of different satellites is different, their phases of revolution will preserve 
the same difference, and the system of satellites will still be arranged in five 
undulations, only the undulations will be propagated round the ring in the 
direction opposite to that of the revolution of the satellites. 

To understand the motion better, let us conceive the centres of the orbits 
of the satellites to be arranged in a straight line instead of a circle, as in 
fig. 10. Each satellite is here represented in a different phase of its orbit, so 
that as we pass from one to another from left to right, we find the position 
of the satellite in its orbit altering in the direction opposite to that of the 
hands of a watch. The satellites all lie in a trochoidal curve, indicated by 
the line through them in the figure. Now conceive every satellite to move in 
its orbit through a certain angle in the direction of the arrows. The satellites 
will then lie in the dotted line, the form of which is the same as that of 
the former curve, only shifted in the direction of the large arrow. It appears, 
therefore, that as the satellites revolve, the undulation travels, so that any 
part of it reaches successively each satellite as it comes into the same phase 
of rotation. It therefore travels from those satellites which are most advanced 
in phase to those which are less so, and passes over a complete wave-length 
in the time of one revolution of a satellite. 

Now if the satellites be arranged as in fig. 8, where each is more advanced 
in phase as we go round the ring in the direction of rotation, the wave will 
travel in the direction opposite to that of rotation, but if they are arranged 
as in fig. 12, where each satellite is less advanced in phase as we go round 
the ring, the wave will travel in the direction of rotation. Fig. 8 represents 
the first series of waves where m = 5, and fig. 12 represents the fourth series 
where m = 7. By arranging the satellites in their sockets before starting, we 
might make w equal to any whole number, from 1 to 18. If we chose any 
number above 18 the result would be the same as if we had taken a number 
as much below 18 and changed the arrangement from the first wave to the 
fourth. 



ON THE STABILITY OF THE MOTION OF SATURN S RINGS. 365 

In this way we can exhibit the motions of the satellites in the first and 
fourth waves. In reality they ought to move in ellipses, the major axes being 
twice the minor, whereas in the machine they move in circles : but the character 
of the motion is the same, though the form of the orbit is diflferent. 

We may now show these motions of the satellites among each other, com- 
bined with the motion of rotation of the whole ring. For this purpose we 
put in the pin P, so as to prevent the crank axle from turning, and take 
out the pin ^ so as to allow the wheel R to turn. If we then turn the 
wheel T, all the small cranks will remain parallel to the fixed crank, and the 
wheel R will revolve at the same rate as T. The arm of each satellite will 
continue parallel to itself during the motion, so that the satellite will describe 
a circle whose centre is at a distance from the centre of R, equal to the arm 
of the satellite, and measured in the same direction. In our theory of real 
satellites, each moves in an ellipse, having the central body in its focus, but 
this motion in an eccentric circle is sufficiently near for illustration. The 
motion of the waves relative to the ring is the same as before. The waves 
of the first kind travel faster than the ring itself, and overtake the satellites, 
those of the fourth kind travel slower, and are overtaken by them. 

In fig. 11 we have an exaggerated representation of a ring of twelve satel- 
lites afiected by a wave of the fourth kind where m = 2. The satellites here lie in 
an eUipse at any given instant, and as each moves round in its circle about 
its mean position, the ellipse also moves round in the same direction with half 
their angular velocity. In the figure the dotted line represents the position of 
the ellipse when each satellite has moved forward into the position represented 
by a dot. 

Fig. 13 represents a wave of the first kind where m = 2. The satellites at 
any instant lie in an epitrochoid, which, as the satellites revolve about their 
mean positions, revolves in the opposite direction with half their angular velocity, 
so that when the satellites come into the positions represented by the dots, 
the curve in which they lie turns round in the opposite direction and forms the 
dotted curve. 

In fig. 9 we have the same case as in fig. 13, only that the absolute orbits 
of the satellites in space are given, instead of their orbits about their mean 
positions in the ring. Here each moves about the central body in an eccentric 



366 ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 

circle, which in strictness ought to be an ellipse not differing much from the 
circle. 

As the satellites move in their orbits in the direction of the arrows, the 
curve which they form revolves in the same direction with a velocity 1^ times 
that of the ring. 

By considering these figures, and still more by watching the actual motion 
of the ivory balls in the model, we may form a distinct notion of the motions 
of the particles of a discontinuous ring, although the motions of the model are 
circular and not elliptic. The model, represented on a scale of one-third in figs. 
7 and 8, was made in brass by Messrs. Smith and Ramage of Aberdeen. 

We are now able to understand the mechanical principle, on account of 
which a massive central body is enabled to govern a numerous assemblage of 
satellites, and to space them out into a regular ring; while a smaller central 
body would allow disturbances to arise among the individual satelHtes, and 
collisions to take place. 

When we calculated the attractions among the satellites composing the ring, 
we found that if any satellite be displaced tangentially, the resultant attraction 
will draw it away from its mean position, for the attraction of the satellites it 
approaches will increase, while that of those it recedes from will diminish, so that 
its equilibrium when in the mean position is unstable with respect to tangential 
displacements ; and therefore, since every satellite of the ring is statically unstable 
between its neighbours, the slightest disturbance would tend to produce coUisions 
among the satellites, and to break up the ring into groups of conglomerated 
sateUites- 

But if we consider the dynamics of the problem, we shall find that this 
effect need not necessarily take place, and that this very force which tends 
towards destruction may become the condition of the preservation of the ring. 
Suppose the whole ring to be revolving round a central body, and that one 
satellite gets in advance of its mean position. It will then be attracted forwards, 
its path will become less concave towards the attracting body, so that its distance 
from that body will increase. At this increased distance its angular velocity 
will be less, so that instead of overtaking those in front, it may by this means 
be made to fall back to its original position. Whether it does so or not must 
depend on the actual values of the attractive forces and on the angular velocity 
of the ring. When the angular velocity is great and the attractive forces small, 



ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 367 

the compensating process will go on vigorously, and the ring wiU be preserved. 
When the angular velocity is small and the attractive forces of the ring great, 
the dynamical effect wiU not compensate for the disturbing action of the forces 
and the ring ^vill be destroyed. 

If the satellite, instead of being displaced forwards, had been originally 
behind its mean position in the ring, the forces would have pulled it backwards, 
its path would have become more concave towards the centre, its distance from 
the centre would diminish, its angular velocity would increase, and it would 
gain upon the rest of the ring till it got in front of its mean position. This 
effect is of course dependent on the very same conditions as in the former case, 
and the actual effect on a disturbed satellite would be to make it describe an 
orbit about its mean position in the ring, so that if in advance of its mean 
position, it first recedes from the centre, then falls behind its mean position in 
the ring, then approaches the centre within the mean distance, then advances 
beyond its mean position, and, lastly, recedes from the centre till it reaches its 
starting-point, after which the process is repeated indefinitely, the orbit being 
always described in the direction opposite to that of the revolution of the 
ring. 

We now understand what would happen to a disturbed satellite, if all the 
others were preserved from disturbance. But, since all the satellites are equally 
free, the motion of one will produce changes in the forces acting on the rest, 
and this will set them in motion, and this motion will be propagated from one 
satellite to another round the ring. Now propagated disturbances constitute 
waves, and all waves, however complicated, may be reduced to combinations of 
simple and regular waves; and therefore all the disturbances of the ring may 
be considered as the resultant of many series of waves, of different lengths, and 
travelling with different velocities. The investigation of the relation between 
the length and velocity of these waves forms the essential part of the problem, 
after which we have only to split up the original disturbance into its simple 
elements, to calculate the effect of each of these separately, and then to combine 
the results. The solution thus obtained will be perfectly general, and quite 
independent of the particular form of the ring, whether regular or irregular at 
starting. § 14. 

We next investigated the effect upon the ring of an external disturbing 
force. Having split up the disturbing force into components of the same type 



368 ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 

with the waves of the ring (an operation which is always possible), we found 
that each term of the disturbing force generates a " forced wave " travelling with 
its own angular velocity. The magnitude of the forced wave depends not only 
on that of the disturbing force, but on the angular velocity with which the dis- 
turbance travels round the ring, being greater in proportion as this velocity 
more nearly coincides with that of one of the "free waves" of the ring, "We 
also found that the displacement of the satellites was sometimes in the direction 
of the disturbing force, and sometimes in the opposite direction, according to 
the relative position of the forced wave among the four natural ones, producing 
in the one case positive, and in the other negative forced waves. In treating 
the problem generally, we must determine the forced waves belonging to every 
term of the disturbing force, and combine these with such a system of free 
waves as shall reproduce the initial state of the ring. The subsequent motion 
of the rmg is that which would result from the free waves and forced waves 
together. The most important class of forced waves are those which are pro- 
duced by waves in neighbouring rings. § 15. 

We concluded the theory of a ring of satellites by tracing the process by 
which the ring would be destroyed if the conditions of stability were not 
fulfilled. We found two cases of instability, depending on the nature of the 
tangential force due to tangential displacement. If this force be in the direction 
opposite to the displacement, that is, if the parts of the ring are statically 
stable, the ring will be destroyed, the irregularities becoming larger and larger 
mthout being propagated round the ring. When the tangential force is in the 
direction of the tangential displacement, if it is below a certain value, the 
disturbances will be propagated round the ring without becoming larger, and 
we have the case of stability treated of at large. If the force exceed this value, 
the disturbances will still travel round the ring, but they will increase in ampli- 
tude continually till the ring falls into confusion. § 18. 

We then proceeded to extend our method to the case of rings of different 
constitutions. The first case was that of a ring of satellites of unequal size. 
If the central body be of suflScient mass, such a ring will be spaced out, so that 
the larger satellites will be at wider intervals than the smaller ones, and the 
waves of disturbance will be propagated as before, except that there may be 
reflected waves when a wave reaches a part of the ring where there is a change 
in the average size of the satellites. § 19. 



ON THE STABILITY OF THE MOTION OF SATURN S RINGS. 369 

The next case was that of an annular cloud of meteoric stones, revolving 
uniformly about the planet. The avercige density of the space through which 
these small bodies are scattered will vary with every irregularity of the motion, 
and this variation of density will produce variations in the forces acting upon 
the other parts of the cloud, and so disturbances will be propagated in this 
ring, as in a ring of a finite number of satellites. The condition that such a 
ring should be free from destructive oscillations is, that the density of the 
planet should be more than three hundred times that of the ring. This would 
make the ring much rarer than common air, as regards its average density, 
though the density of the particles of which it is composed may be great. 
Comparing this result with Laplace's minimum density of a ring revolving as 
a whole, we find that such a ring cannot revolve as a whole, but that the inner 
parts must have a greater angular velocity than the outer parts. § 20. 

We next took up the case of a flattened ring, composed of incompressible 
fluid, and moving with uniform angular velocity. The internal forces here arise 
partly from attraction and partly from fluid pressure. We began by taking the 
case of an intinite stratum of fluid affected by regular waves, and found the accurate 
values of the forces in this case. For long waves the resultant force Is in the 
same direction as the displacement, reaching a maximum for waves whose 
length is about ten times the thickness of the stratum. For waves about five 
times as long as the stratum is thick there is no resultant force, and for shorter 
waves the force is in the opposite direction to the displacement. § 23. 

Applying these results to the case of the ring, we find that it will be 
destroyed by the long waves unless the fluid is less than -^ of the density of 
the planet, and that in all cases the short waves will break up the ring into 
small satellites. 

Passing to the case of narroiv rings, we should find a somewhat larger 
maximum density, but we should still find that very short waves produce forces 
in the direction opposite to the displacement, and that therefore, as already 
explained (page 333), these short undulations would increase in magnitude without 
being propagated along the ring, till they had broken up the fluid filament into 
drops. These drops may or may not fulfil the condition formerly given for the 
stability of a ring of equal satellites. If they fulfil it, they will move as a 
permanent ring. If they do not, short waves will arise and be propagated among 
the satellites, with ever increasing magnitude, till a sufficient number of drops 



370 ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 

have been brought into collision, so as to unite and form a smaller number of 
larger drops, which may be capable of revolving as a permanent ring. 

We have already investigated the disturbances produced by an external 
force independent of the ring ; but the special case of the mutual perturbations 
of two concentric rings is considerably more complex, because the existence of a 
double system of waves changes the character of both, and the waves produced 
react on those that produced them. 

We determined the attraction of a ring upon a particle of a concentric 
ring, first, when both rings are in their undisturbed state ; secondly, when the 
particle is disturbed ; and, thirdly, when the attracting ring is disturbed by a 
series of waves. § 26. 

We then formed the equations of motion of one of the rings, taking in the 
disturbing forces arising from the existence of a wave in the other ring, and 
found the small variation of the velocity of a wave in the first ring as dependent 
on the magnitude of the wave in the second ring, which travels with it. § 27. 

The forced wave in the second ring must have the same absolute angular 
velocity as the free wave of the first which produces it, but this velocity of 
the free wave is slightly altered by the reaction of the forced wave upon it. 
We find that if a free wave of the first ring has an absolute angular velocity 
not very different from that of a free wave of the second ring, then if both 
fi:ee waves be of even orders (that is, of the second or fourth varieties of waves), 
or both of odd orders (that is, of the first or third), then the swifter of the 
two free waves has its velocity increased by the forced wave which it produces, 
and the slower free wave is rendered still slower by its forced wave ; and even 
when the two free waves have the same angular velocity, their mutual action 
will make them both split into two, one wave in each ring travelling faster, 
and the other wave in each ring travelling slower, than the rate with which 
they would move if they had not acted on each other. 

But if one of the free waves be of an even order and the other of an odd 
order, the swifter free wave will travel slower, and the slower free wave will 
travel swifter, on account of the reaction of their respective forced waves. If 
the two free waves have naturally a certain small difference of velocities, they 
will be made to travel together, but if the difference is less than this, they 
will again split into two pairs of waves, one pair continually increasing in 



ON THE STABILITY OF THE MOTION OF SATURN S RINGS. 371 

magnitude without limit, and the other continually diminishing, 30 that one 
of the waves in each ring will increase in violence till it has thrown the ring 
into a state of confusion. 

There are four cases in which this may happen. The first wave of the 
outer ring may conspire with the second or the fourth of the inner ring, the 
second of the outer with the third of the inner, or the third of the outer with 
the fourth of the inner. That two rings may revolve permanently, their distances 
must be arranged so that none of these conspiracies may arise between odd 
and even waves, whatever be the value of m. The number of conditions to 
be fulfilled is therefore very great, especially when the rings are near together 
and have nearly the same angular velocity, because then there are a greater 
number of dangerous values of m to be provided for. 

In the case of a large number of concentric rings, the stability of each pair 
must be investigated separately, and if in the case of any two, whether con- 
secutive rings or not, there are a pair of conspiring waves, those two rings will 
be agitated more and more, till waves of that kind are rendered impossible by 
the breaking up of those rings into some different arrangement. The presence 
of the other rings cannot prevent the mutual destruction of any pair which 
bear such relations to each other. 

It appears, therefore, that in a system of many concentric rings there will 
be continually new cases of mutual interference between different pairs of rings. 
The forces which excite these disturbances being very small, they will be slow 
of growth, and it is possible that by the irregularities of each of the rings the 
waves may be so broken and confused (see § 19), as to be incapable of mounting 
up to the height at which they would begin to destroy the arrangement of the 
ring. In this way it may be conceived to be possible that the gradual dis- 
arrangement of the system may be retarded or indefinitely postponed. 

But supposing that these waves mount up so as to produce collisions among 
the particles, then we may deduce the result upon the system from general 
dynamical principles. There will be a tendency among the exterior rings to 
remove further from the planet, and among the interior rings to approach the 
planet, and this either by the extreme interior and exterior rings diverging 
from each other, or by intermediate parts of the system moving away from the 
mean ring. If the interior rings are observed to approach the planet, while it 

47—2 



372 ON THE STABILITY OF THE MOTION OF SATURN S RINGS. 

is known that none of the other rings have expanded, then the cause of the 
chancre cannot be the mutual action of the parts of the system, but the resistance 
of some medium in which the rings revolve. § Si- 
There is another cause which would gradually act upon a broad fluid ring 
of which the parts revolve each with the angular velocity due to its distance 
from the planet, namely, the internal friction produced by the slipping of the 
concentric rings with different angular velocities. It appears, however (§ 33), 
that the effect of fluid friction would be insensible if the motion were regular. 

Let us now gather together the conclusions we have been able to draw 
from the mathematical theory of various kinds of conceivable rings. 

We found that the stability of the motion of a solid ring depended on 
so delicate an adjustment, and at the same time so unsymmetrieal a distribution 
of mass, that even if the exact condition were fulfilled, it could scarcely last 
long, and if it did, the immense preponderance of one side of the ring would 
be easily observed, contrary to experience. These considerations, with others 
derived from the mechanical structure of so vast a body, compel us to abandon 
any theory of solid rings. 

We next examined the motion of a ring of equal satellites, and found that 
if the mass of the planet is sufficient, any disturbances produced in the arrange- 
ment of the ring will be propagated round it in the form of waves, and will not 
introduce dangerous confusion. If the satellites are unequal, the propagation of 
the waves will no longer be regular, but disturbances of the ring will in this, 
as in the former case, produce only waves, and not growing confusion. Sup- 
posing the ring to consist, not of a single row of large satellites, but of a cloud 
of evenly distributed unconnected particles, we found that such a cloud must 
have a very small density in order to be permanent, and that this is inconsistent 
with its outer and inner parts moving with the same angular velocity. Supposing 
the ring to be fluid and continuous, we found that it will be necessarily broken 
up into small portions. 

We conclude, therefore, that the rings must consist of disconnected particles ; 
these may be either solid or liquid, but they must be independent. The entire 
system of rings must therefore consist either of a series of many concentric rings, 
each moving with its own velocity, and having its own systems of waves, or else 
of a confused multitude of revolving particles, not arranged in rings, and 
continually coming into collision with each other. 



ON THE STABILITY OF THE MOTION OF rfATUKN S RINGS. 373 

Taking the first case, we tbund that in an indefinite number of possible 
cases the mutual perturbations of two rings, stable in themselves, might mount 
up in time to a destructive magnitude, and that such cases must continually 
occur in an extensive system like that of Saturn, the only retarding cause being 
the possible irregularity of the rings. 

The result of long-continued disturbance was found to be the spreading 
out of the rings in breadth, the outer rings pressing outwards, while the inner 
rings press inwards. 

The final result, therefore, of the mechanical theory is, that the only system 
of rings which can exist is one composed of an indefinite number of unconnected 
particles, revolving round the planet with different velocities according to their 
respective distances. These particles may be arranged in series of narrow rings, 
or they may move through each other irregularly. In the first case the destruc- 
tion of the system will be very slow, in the second case it will be more rapid, 
but there may be a tendency towards an arrangement in narrow rings, which 
may retard the process. 

We are not able to ascertain by observation the constitution of the two 
outer divisions of the system of rings, but the inner ring is certainly transparent, 
for the limb of Saturn has been observed through it. It is also certain, that 
though the space occupied by the ring is transparent, it is not through the 
material parts of it that Saturn was seen, for his limb was observed without 
distortion ; which shows that there was no refraction, and therefore that the 
rays did not pass through a medium at all, but between the solid or liquid 
particles of which the ring is composed. Here then we have an optical argument 
in favour of the theory of independent particles as the material of the rings. 
The two outer rings may be of the same nature, but not so exceedingly rare 
that a ray of light can pass through their whole thickness without encounterino^ 
one of the particles. 

Finally, the two outer rings have been observed for 200 years, and it appears, 
from the careful analysis of all the observations by M. Struve, that the second 
ring is broader than when first observed, and that its inner edge is nearer the 
planet than formerly. The inner ring also is suspected to be approaching the 
planet ever since its discovery in 1850. These appearances seem to indicate 
the same slow progress of the rings towards separation which we found to be 
the result of theory, and the remark, that the inner edge of the inner ring is 



374 ON THE STABILITY OF THE MOTION OF SATURN S RINGS. 

most distinct, seems to indicate that the approach towards the planet is less 
rapid near the edge, as we had reason to conjecture. As to the apparent 
unchangeableness of the exterior diameter of the outer ring, we must remember 
that the outer rings are certainly far more dense than the inner one, and that 
a small change in the outer rings must balance a great change in the inner 
one. It is possible, however, that some of the observed changes may be due 
to the existence of a resisting medium. If the changes already suspected should 
be confirmed by repeated observations with the same instruments, it will be 
worth while to investigate more carefully whether Saturn's Rings are permanent 
or transitionary elements of the Solar System, and whether in that part of 
the heavens we see celestial immutability, or terrestrial corruption and generation, 
and the old order giving place to new before our own eyes. 



APPENDIX. 

On the Stability of the Steady Motion of a Rigid Body about a Fixed Centre of Force. 
By Peofessor W. Thomson {communicated in a letter). 

The body will be supposed to be symmetrical on the two sides of a certain plane 
containing the centre of force, and no motion except that of parts of the body parallel 
to the plane will be considered. Taking it as the plane of construction, let G (fig. 14) 
be the centre of gravity of the body, and a point at which the resultant attraction of 
the body is in the line OG towards G. Then if the body be placed with coinciding 
with the centre of force, and set in a state of rotation about that point as an axis, with 

an angular velocity equal to A/Ajr. (where / denotes the attraction of the body on a 

unit of matter at 0, S the amount of matter in the central body, M the mass of the 
revolving body, and a the distance OG), it will continue, provided it be perfectly undis- 
turbed, to revolve uniformly at this rate, and the attraction Sf on the moving body will 
be constantly balanced by the centrifugal force oi'aM of its motion. 

Let us now suppose the motion to be slightly disturbed, and let it be required to 
investigate the consequences. Let X, S, Y, be rectangular axes of reference revolving 
uniformly with the angular velocity (o, round S, the fixed attracting point. Let x, y, be 
the co-ordinates of G with reference to these axes, and let XS, YS denote the components 



ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 375 

of the whole force of attraction of S on the rigid body. Then since this force is in the 
line through S, its moment round G is 

SYx-SXy; 
the components of the forces on the moving body being reckoned as positive when they 
tend to diminish x and y respectively. Hence if k denote the radius of gyration of the 
body round G, and if <f> denote the angle which OG makes with SX {i.e. the angle GOK), 
the equations of motion are, 

In the first place we see that one integral of these equations is 

This is the "equation of angidar momentum." 

In considering whether the motion round S with velocity co when coincides with 
-S' is stable or unstable, we must find whether every possible motion with the same 
" angular momentum " round S is such that it will never bring to more than an infinitely 
small distance from S : that is to say, we must find whether, for every possible solution 
in which H = M {ct" + k"") o), and for which the co-ordinates of are infinitely small at one 
time, these co-ordinates remain infinitely small. Let these values at time t be denoted 
thus: 8^ = ^, and NO='rj; let OG be at first infinitely nearly parallel to OX, i.e. let <f> 
be infinitely small (the full solution will tell us whether or not <f) remains infinitely small) ; 
then, as long as <f) is infinitely small, we have 

x = a+ ^, y = v + ^<^> 
and the equations of motion have the forms 

31 






and we may write the equation of angular momentum instead of the third equation. 

If now we suppose f and rj to be infinitely small, the last of these equations becomes 
{a' + k^)f^+2a>a^+af^=0 (a). 



376 ON THE STABILITY OF THE MOTION OF SATUKN S RINGS. 

If p and q denote the components parallel and perpendicular to OG of the attraction 
of the body on a unit of matter at S, we have 

X = pco?,^-q?,m4> = p, and F=psin^ + 5^003 ^=j3</> 4-^, 
since q and ^ are each infinitely small ; and if we put V= potential at S, and 

then p =/- a| - 777, q = -0v- 7^. 

If we make these substitutions for X and Y, and take into account that 

.f=co'a^ (*). 

the first and second equations of motion become 

g_2.^_„.f_2„af4(.f+„)=0 (0), 

A,2„|_„., + „^4(^,+,« = W. 

Combining equations (a), (c), and (tf), by the same method as that adopted in the text, 
we find that the differential equation in ^, 7), or </>, is of the form 



d*u ^d^u ^ 



where A = A;', 



C = a>* (A;* - 3a*) + «" -^ {{a* + ^*) (a + yS) - 4a»y8} + {a' + Fj^^, (a'yS - 7). 

In comparing this result with that obtained in the Essay, we must put 

r^ for a, 

R for M, 

B+S for S, 

L for o, 

Nt: for y8, 

Mr^ for 7. 



Tv^ 7 



Fi^ Z 



VOL. L PLATE V. 




Tig. ^. 



Fi^ 6. 



VOL. L PLATE V, 




[From the Philosophical Magazine for January and July, I860.] 



XX. Illustrations of the Dynamical Theory of Gases*. 

PART L 

On the Motions and Collisions of Perfectly Elastic Spheres. 

So many of the properties of matter, especially when in the gaseous form, 
can be deduced from the hypothesis that their minute parts are in rapid motion, 
the velocity increasing with the temperature, that the precise nature of this 
motion becomes a subject of rational curiosity. Daniel Bemouilli, Herapath, 
Joule, Kronig, Clausius, &c. have shewn that the relations between pressure, 
temperature, and density in a perfect gas can be explained by supposing the 
particles to move with uniform velocity in straight lines, striking against the 
sides of the containing vessel and thus producing pressure. It is not necessary 
to suppose each particle to travel to any great distance in the same straight 
line ; for the effect in producing pressure \vill be the same if the particles 
strike against each other ; so that the straight line described may be very short . 
M. Clausius has determined the mean length of path in terms of the average 
distance of the particles, and the distance between the centres of two particles 
when collision takes place. We have at present no means of ascertaining either 
of these distances ; but certain phenomena, such as the internal friction of gases, 
the conduction of heat through a gas, and the diffusion of one gas through 
another, seem to indicate the possibility of determining accurately the mean 
length of path which a particle describes between two successive collisions. In 
order to lay the foundation of such investigations on strict mechanical principles, 
I shall demonstrate the laws of motion of an indefinite number of small, hard, 
and perfectly elastic spheres acting on one another only during impact. 

* Read at the Meeting of the British Association at Aberdeen, Sei)tember 21, 1859. 
VOL. I. 48 



378 ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES. 

If the properties of such a system of bodies are found to correspond to 
those of gases, an important physical analogy will be established, which may 
lead to more accurate knowledge of the properties of matter. If experiments 
on gases are inconsistent with the hypothesis of these propositions, then our 
theory, though consistent w^th itself, is proved to be incapable of explaining 
the phenomena of gases. In either case it is necessary to follow out the 
consequences of the hypothesis. 

Instead of saying that the particles are hard, spherical, and elastic, we may 
if we please say that the particles are centres of force, of which the action is 
insensible except at a certain small distance, when it suddenly appears as a 
repulsive force of very great intensity. It is evident that either assumption 
will lead to the same results. For the sake of avoiding the repetition of a 
long phrase about these repulsive forces, I shall proceed upon the assumption 
of perfectly elastic spherical bodies. If we suppose those aggregate molecules 
which move together to have a bounding surface which is not spherical, then 
the rotatory motion of the system will store up a certain proportion of the 
whole vis viva, as has been shewn by Clausius, and in this way we may 
accoimt for the value of the specific heat being greater than on the more 
simple hypothesis. 

On the Motion and Collision of Perfectly Elastic Spheres. 

Prop. I. Two spheres moving in opposite directions with velocities* inversely 
us their masses strike one another; to determine their motions after impact. 

Let P and Q be the position of the centres at 
impact; AP, BQ the directions and magnitudes of ^-V at 

the velocities before impact; Pa, Qh the same after ^^^^^^^^ — j^ 

impact; then, resolving the velocities parallel and per- ^ 
pendicular to PQ the line of centres, we find that 
tlie velocities parallel to the line of centres are exactly 
reversed, while those perpendicular to that line are 
luichanged. Compounding these velocities again, we find that the velocity of 
each ball is the same before and after impact, and that the directions before 
and after impact lie in the same plane with the line of centres, and make equal 
angles with it. 




ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES. 379 

Prop. 11. To find the probability of the direction of the velocity after 
impact lying between given limits. 

In order that a collision may take place, the line of motion of one of the 
balls must pass the centre of the other at a distance less than the sum of 
their radii ; that is, it must pass through a circle whose centre is that of the 
other ball, and radius (s) the sum of the radii of the balls. Within this circle 
every position is equally probable, and therefore the probability of the distance 
from the centre being between r and r + dr is 

2rdr 
~7~' 

Now let <f> be the angle A Pa between the original direction and the directioii 
after impact, then APN=^<f>, and 7- = 5 sin ^<^, and the probabihty becomes 

^ sin 6d^. 
The area of a spherical zone between the angles of polar distance <j> and <f) + d<f) is 

27r sin (f)d<f> ; 

therefore if a> be any small area on the surface of a sphere, radius unity, the 
probability of the direction of rebound passing through this area is 

to 

4:ir * 

so that the probability is independent of ^, that is, all directions of rebound 
are equally likely. 

Prop. III. Given the direction and magnitude of the velocities of two 
spheres before impact, and the line of centres at impact ; to find the velocities 
after impact. 

Let OA, OB represent the velocities before impact, so that if there had been 
no action between the bodies they would 
have been at A and B at the end of a 
second. Join AB, and let G be their centre 
of gravity, the position of which is not 
affected by their mutual action. Draw GN 
parallel to the line of centres at impact (not 
necessarily in the plane AOB). Draw aGh 

in the plane AGN, making NGa = NGA, and Ga=GA and Gb = GB; then by 

48—2 




380 ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES. 

Prop. I. Ga and Gh will be the velocities relative to G ; and compounding 
these with OG, we have Oa and Oh for the true velocities after impact. 

By Prop. 11. all directions of the Une aGh are equally probable. It appears 
therefore that the velocity after impact is compounded of the velocity of the 
centre of gravity, and of a velocity equal to the velocity of the sphere relative 
to the centre of gravity, which may with equal probability be in any direction 
whatever. 

If a great many equal spherical particles were in motion in a perfectly 
elastic vessel, collisions would take place among the particles, and their velocities 
would be altered at every collision; so that after a certain time the vis viva 
will be divided among the particles according to some regular law, the average 
number of particles whose velocity lies between certain Umits being ascertainable, 
though the velocity of each particle changes at every colUsion. 

Prop. IV. To find the average number of particles whose velocities he 
between given limits, after a great number of collisions among a great number 
of equal particles. 

Let N be the whole number of particles. Let x, y, z be the components 
of the velocity of each particle in three rectangular directions, and let the number 
of particles for which x lies between x and x-hdx, be Nf{x)dx, where f{x) is 
a function of x to be determined. 

The number of particles for which y lies between y and y + dy wUl be 
Nf{y)dy; and the number for which z Hes between z and z + dz will be Nf(z)dz, 
where / always stands for the same function. 

Now the existence of the velocity x does not in any way affect that of 
the velocities y or z, since these are all at right angles to each other and 
independent, so that the number of particles whose velocity lies between x and 
x + dx, and also between y and y-{-dy, and also between z and z + dz, is 

If we suppose the N particles to start from the origin at the same instant, 
then this wil) be the number in the element of volume (dxdydz) after unit of 
time, and the number referred to unit of volume will be 



ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES. 381 

But the directions of the coordinates are perfectly arbitrary, and therefore this 
number must depend on the distance from the origin alone, that is 

f{x)f(y)f(z) = ^{^+y' + z% 

Solving this functional equation, we find 

f{x) = Ce^'', (^M = CV. 

If we make A positive, the number of particles will increase with the 
velocity, and we should find the whole number of particles infinite. We there- 
fore make A negative and equal to — „ , so that the number between x and 

x + dx is 

NCe'^'dx. 

Integrating from a:=— <» toa;=-foo,we find the whole number of particles, 

aVTT 

1 -?: 

f[x) is therefore /-e " . 

Whence we may draw the following conclusions : — 

1st. The number of particles whose velocity, resolved in a certain direction, 
lies between x and x + dx is 

N^i'^'dx (1). 

2nd. The number whose actual velocity lies between v and v + dv is 

]Sf-^^^e~^'dv (2). 

3rd. To find the mean value of v, add the velocities of all the particles 
together and divide by the number of particles ; the result is 

mean velocity = -p- (3). 

Vtt 

4th. To find the mean value of v; add all the values together and 

divide by N, 

mean value of t;' = |a- (4). 

This is greater than the square of the mean velocity, as it ought to be. 



382 ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES. 

It appears from this proposition that the velocities are distributed among 
the particles according to the same law as the errors are distributed among 
the observations in the theory of the " method of least squares." The velocities 
i-ange from to oo , but the number of those having great velocities is com- 
paratively small. In addition to these velocities, which are in all directions 
equally, there may be a general motion of translation of the entire system of 
particles which must be compounded with the motion of the particles relatively 
to one another. We may call the one the motion of translation, and the other 
the motion of agitation. 

Prop. V. Two systems of particles move each according to the law stated 
in Prop. IV. ; to find the number of pairs of particles, one of each system, 
whose relative velocity lies between given limits. 

Let there be N particles of the first system, and N' of the second, then 
NN' is the whole number of such pairs. Let us consider the velocities in the 
direction of x only ; then by Prop. IV. the number of the first kind, whose 
velocities are between x and x + dx, is 

1 -^ 
N — j=e '^ dx. 

aV-Tr 

The number of the second kind, whose velocity is between x + y and x + y + dy, is 

1 (i±vl 
N' — 7= e ^ dy, 

where fi is the value of a for the second system. 

The number of pairs which fulfil both conditions is 

NN'^e'^^'^' dxdy. 
apir 

Now X may have any value from — qo to +cx> consistently with the difference 
of velocities being between y and y + dy; therefore integrating between these 
limits, we find 

^^'7^^^"'^''^ ^'^ 

for the whole number of pairs whose difference of velocity lies between y and 
y + dy. 



ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES. ;}83 

This expression, which is of the same form with (1) if we put XN' for 
X, a' + ^ for a', and y for x, shews that the distribution of relative velocities 
is regulated by the same law as that of the velocities themselves, and that 
the mean relative velocity is the square root of the sum of the squares of tlie 
mean velocities of the two systems. 

Since the direction of motion of every particle in one of the systems may 
be reversed without changing the distribution of velocities, it follows that the 
velocities compounded of the velocities of two particles, one in each system, .-irr 
distributed according to the same formula (5) as the relative velocities. 

Prop. VI. Two systems of particles move in the same vessel ; to prove 
that the mean vis viva of each particle will become the same in the two 
systems. 

Let P be the mass of each particle of the first system, Q that of each 
particle of the second. Let p, q be the mean veloci- 
ties in the two systems before impact, and let p', (( 
be the mean velocities after one impact. Let OA = p 
and OB = q, and let AOB be a right angle; then, by 
Prop, v., AB will be the mean relative velocity, OG will 
be the mean velocity of the centre of gravity ; and drawing 
aGh at right angles to OG, and making aG = AG and 
bG = BG, then Oa will be the mean velocity of P after 
impact, compounded of OG and Ga, and Ob will be that of Q after impact. 





^~ P+Q ' 

therefore p' = Oa = ^!^^±^Ipl±^: , 

^ P + Q 

and q' = Ob = ^-^M±S±El±W, 

P+Q 

and Pp"-Qq" = {^)\Pp'-Qq') C^). 



384 ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES. 

It appears therefore tKat the quantity Pp' — Qq^ is diminished at every impact 
in the same ratio, so that after many impacts it will vanish, and then 

Now the mean vis viva is f Pa'' = -^ Pp* for P, and ^ Qq^ for Q ; and it is 

8 8 

manifest that these quantities will be equal when Pp^ = Qq^. 

If any number of different kinds of particles, having masses P, Q, R and 
velocities jp, q, r respectively, move in the same vessel, then after many impacts 

Pf^Q^ = m^, &c (7). 

Prop. VII. A particle moves with velocity r relatively to a number of 
particles of which there are N in imit of volume ; to find the number of these 
which it approaches within a distance 5 in unit of time. 

If we describe a tubular surface of which the axis is the path of the 
particle, and the radius the distance s, the content of this surface generated 
in unit of time will be irrs^, and the number of particles included in it will be 

Nirrs' (8), 

which is the number of particles to which the moving particle approaches within 
a distance s. 

Prop. VIII. A particle moves with velocity v in a system moving according 
to the law of Prop. IV.; to find the number of particles which have a velocity 
relative to the moving particle between r and r + dr. 

Let u be the actual velocity of a particle of the system, v that of the 
original particle, and r their relative velocity^ and 6 the angle between v and r, 
then 

u^z=v^ + 7^ — 2vr cos 0. 

If we suppose, as in Prop. IV., all the particles to start from the origin, at 
once, then after imit of time the "density" or number of particles to unit of 
volume at distance u will be 

1 -^ 
aM 



ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES. 385 

From this we have to deduce the number of particles in a shell whose centre 
is at distance v, radius = r, and thickness = dr, 

^-n=l{^ *• -« *^ }^^' (9)> 

which is the number required. 

CoR. It is evident that if we integrate this expression from r = to 
/• = oo , we ought to get the whole number of particles = iV, whence the following 
mathematical result, 

dx.x{e »' —e~ »' ) = V77-aa (lO). 

Prop. IX. Two sets of particles move as in Prop. V.; to find the number 
of pairs which approach within a distance s in unit of time. 

The number of the second kind which have a velocity between v and v + dv ia 

4 -^ 

The number of the first kind whose velocity relative to these is between r 
and ri-dr is 

iV — = - (e »' -e »* )dr = n, 
and the number of pairs which approach within distance 5 in unit of time is 

4 t. _ ("-»•)* (o^-r)* 

^NN' -^.s'r've ^ {e »' -e" «" \drdv. 
By the last proposition we are able to integrate with respect to v, and get 

Integrating this again from r = to r = oo , 

2NN' J^ J'^FT^s' (11) 

is the number of collisions in unit of time which take place in unit of volume 

between particles of difierent kinds, s being the distance of centres at collision. 

vol. I. 49 



386 ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES. 

The number of collisions between two particles of the first kind, 5, being the 
striking distance, is 

and for the second system it is 

The mean velocities in the two systems are -7= and -^ ; so that if l^ and l^ 

be the mean distances travelled by particles of the first and second systems 
between each collision, then 



ii a 

Prop. X. To find the probability of a particle reaching a given distance 
before striking any other. 

Let us suppose that the probability of a particle being stopped while 
passing through a distance dx, is adx ; that is, if iV particles arrived at a 
distance x, Nadx of them would be stopped before getting to a distance x-^dx. 
Putting this mathematically, 

^=-Na, or N=Ce-'^. 

Putting iV"=l when x = 0, we find e""* for the probability of a particle not 
striking another before it reaches a distance x. 

The mean distance travelled by each particle before striking is - = l. The 

probability of a particle reaching a distance = 7i? without being struck is e"". 
(See a paper by M. Clausius, Philosophical Magazine, February 1859.) 

If all the particles are at rest but one, then the value of a is 

a = Trs'N, 
where s is the distance between the centres at collision, and N is the number 
of particles in unit of volume. If v be the velocity of the moving particle 
relatively to the rest, then the number of collisions in unit of time wiU be 

virs W : 



ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES. 387 

and if V, be the actual velocity, then the number will be r,a ; therefore 



a = -7rsW, 



where v, is the actual velocity of the striking particle, and v its velocity 
relatively to those it strikes. If -y, be the actual velocity of the other particles, 
then V — Jv* + v*. If i\ = i\ , then v = sl2i\ , and 

a = j2TTS*N. 
Note*. M. Clausius makes a = ^Trs^N, 

Prop. XI. In a mixture of particles of two different kinds, to find the 
mean path of each particle. 

Let there be iV, of the first, and N^ of the second in unit of volume. 
Let Si be the distance of centres for a collision between two particles of the 
first set, 5j for the second set, and s for collision between one of each kind. 
Let r, and i\ be the coefficients of velocity, M^, M^ the mass of each particle. 

The probability of a particle M^ not being struck till after reaching a 
distance x, by another particle of the same kind is 

* [In the Philosophical Magazine of 1860, Vol I. pp. 434 — 6 Clausius explains the method by 
which he found his value of the mean relative velocity. It is briefly as follows: If u, v be the 
velocities of two particles their relative velocity is >Ju* + v* - 2uv cos 6 and the mean of this as 
regards direction only, all directions of v being equally probable, is shewn to be 

1 w* , ^ 1 V* , 

f + o — when u<v, and w + ^ — when u> v. 
o V 3 w 

If r = M these expressions coincide. Clausius in applying this result and putting u, v for the 
mean velocities assumes that the mean relative velocity is given by expressions of the same form, 
so that when the mean velocities are each equal to u the mean relative velocity would be ^u. 
This step is, however, open to objection, and in fact if we take the expressions given above for the 
mean velocity, treating u and v as the velocities of two particles which may have any values between 
and 00 , to calculate the mean relative velocity we should proceed as follows : Since the number of 

4 _*! 

particles with velocities between u and w + rfu is N , , tt*g~«' du, the mean relative velocity is 

2 

This expression, when reduced, leads to -j= Ja* + /3', which is the result in the text. Ed.] 

49—2 



388 ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES. 

The probability of not being struck by a particle of the other kind in the same 
distance is 

Therefore the probability of not being struck by any particle before reaching a 
distance x is 

and if k be the mean distance for a particle of the first kind, 

\ = j27rs-N, + 7: ^f^.s^N, (12). 

Similarly, if k be the mean distance for a particle of the second kind, 

l=^/27r5,W, + 7^ /l+^^/W, (13). 

The mean density of the particles of the first kind is N,M, = p„ and that of 
the second NJiI, = p,. If we put 

i =Ap, + Bp,, l = Cp, + Dp, (15), 

^^ C-Wr~< ^ ^ 

Prop. XII. To find the pressure on unit of area of the side of the vessel 
due to the impact of the particles upon it. 

Let iV= number of particles in unit of volume; 
M= mass of each particle ; 
V = velocity of each particle ; 
I = mean path of each particle ; 
then the number of particles in unit of area of a stratum dz thick is 

Ndz (17). 

The number of colHsions of these particles in unit of time is 

Ndz J (18). 



ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES. 389 

The number of particles which after collision reach a distance between nl and 
(n 4- dn) I is 

Njc-^dzdn (19). 



The proportion of these which strike on unit of area at distance z is 

rd — z 



,(20); 



2nl 

the mean velocity of these in the direction of 2 is 

.'4±? (21). 

Multiplying together (19), (20), and (21), and M, we find the momentum at 
impact 

MN-^j,(nn'-z')e-''dzdn. 

Integrating with respect to z from to nl, we get 

^MNi? nt"" dn. 
Integrating with respect to n from to 00 , we get 

for the momentum in the direction of z of the striking particles ; for the 
momentum of the particles after impact is the same, but in the opposite 
direction ; so that the whole pressure on unit of area is twice this quantity, or 

This value of _p is independent of I the length of path. In applying this 
result to the theory of gases, we put MN=p, and v- = 2>h, and then 

which is Boyle and Mariotte's law. By (4) we have 

^'^ = |a^ .-. o: = 2k (23). 

We have seen that, on the hypothesis of elastic particles moving in straight 
lines, the pressure of a gas can be explained by the assumption that the square 
of the velocity is proportional directly to the absolute temperature, and inversely 
to the specific gravity of the gas at constant temperature, so that at the same 



390 ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES. 

r 

pressure and temperature the value of NMif is the same for all gases. But 

we found in Prop. VI. that when two sets of particles communicate agitation 
to one another, the value of Mif is the same in each. From this it appears 
that N, the number of particles in unit of volume, is the same for all gases 
at the same pressure and temperature. This result agrees with the chemical law, 
that equal volumes of gases are chemically equivalent. 

We have next to determine the value of I, the mean length of the path 
of a particle between consecutive collisions. The most direct method of doing 
this depends upon the fact, that when different strata of a gas slide upon 
one another with different velocities, they act upon one another with a tan- 
gential force tending to prevent this sliding, and similar in its results to the 
friction between two solid surfaces sliding over each other in the same way. 
The explanation of gaseous friction, according to our hypothesis, is, that particles 
having the mean velocity of translation belonging to one layer of the gas, pass 
out of it into another layer having a different velocity of translation ; and 
by striking against the particles of the second layer, exert upon it a tangential 
force which constitutes the internal friction of the gas. The whole friction 
between two portions of gas separated by a plane surface, depends upon the 
total action between all the layers on the one side of that surface upon all the 
layers on the other side. 

Prop. XIII. To find the internal friction in a system of moving particles. 

Let the system be divided into layers parallel to the plane of xy, and 
let the motion of translation of each layer be u in the direction of x, and 
let u = A+Bz. We have to consider the mutual action between the layers on 
the positive and negative sides of the plane xy. Let us first determine the 
action between two layers dz and dz\ at distances z and — z' on opposite sides 
of this plane, each unit of area. The number of particles which, starting from 
dz in unit of time, reach a distance between nl and (n-{-dn)l is by (19), 

N J e"** dz dn. 

The number of these which have the ends of their paths in the layer dz' is 

N — -jt e"" dz dz' dn. 

The mean velocity in the direction of x which each of these has before impact 
is A + Bz, and after impact A+Bz'; and its mass is M, so that a mean 



ILLUSTRATIONS OF THE DYNAAIICAL THEORY OF GASES. 391 

momentum =MB{z-z) is communicated by each particle. The whole action due 
to these collisions is therefore 

NMB ^, (z - z) e-** dz dz dn. 

We must first integrate with respect to z' between z' = and z' = z — nl; this 
gives 

^NMB 2^ (nH' -z')e-''dz dn 

for the action between the layer dz and all the layers below the plane xy. 
Then integrate from z = to z = nl, 

^MNBlm'e-'' dn. 
Integrate from n = to n = oo , and we find the whole friction between unit 
of area above and below the plane to be 

where /x is the ordinary coefficient of internal friction, 

-i'^^-iTlS" • ^^^>' 

where p is the density, I the mean length of path of a particle, and v the 

... 2a ^ lYk 

mean velocity v = -j= = 2 J — , 



'=I^V.T (^^)- 



Now Professor Stokes finds by experiments on air. 



J: 



'^ = •116. 

If we suppose n/^ = 930 feet per second for air at 60°, and therefore the mean 
velocity 1^ = 1505 feet per second, then the value of I, the mean distance 
travelled over by a particle between consecutive collisions, =4 47^000 ^^ ^^ ^^ 
inch, and each particle makes 8,077,200,000 collisions per second. 

A remarkable result here presented to us in equation (24), is that if this 
explanation of gaseous friction be true, the coefficient of friction is independent 
of the density. Such a consequence of a mathematical theory is very startling, 
and the only experiment I have met with on the subject does not seem to 
confirm it. We must next compare our theory with what is known of the 
difiusion of gases, and the conduction of heat through a gas. 



392 ILLUSTRATIONS OF THE DYNAMICAI. THEORY OF GASES. 



PART II. 
* On the Process of Diffusion of two or more kinds of moving particles 

AMONG one AI^OTHER. 

We have shewn, in the first part of this paper, that the motions of a 
system of many small elastic particles are of two kinds : one, a general motion 
of translation of the whole system, which may be called the motion in mass; 
and the other a motion of agitation, or molecular motion, in virtue of which 
velocities in all directions are distributed among the particles according to a 
certain law. In the cases we are considering, the collisions are so frequent that 
the law of distribution of the molecular velocities, if disturbed in any way, 
will be re-established in an inappreciably short time; so that the motion will 
always consist of this definite motion of agitation, combined with the general 
motion of translation. 

When two gases are in communication, streams of the two gases might 
run freely in opposite directions, if it were not for the collisions which take 
place between the particles. The rate at which they actually interpenetrate each 
other must be investigated. The diffusion is due partly to the spreading of the 
particles by the molecular agitation, and partly to the actual motion of the 
two opposite currents in mass, produced by the pressure behind, and resisted 

* [The methods and results of this paper have been criticised by Clausius in a memoir published 
in PoggendorflTs Anncden, VoL cxv., and in the Philosophical Magazine, Vol xxiiL His main objec- 
tion is that the various circumstances of the strata, discussed in the paper, have not been sufficiently 
represented in the equations. In particular, if there be a series of strata at different temperatures 
perpendicular to the axis of x, then the proportion of molecules whose directions form with the 
axis of X angles whose cosines lie between /a and /i + <?/x is not \dfj. sa has been assumed by Maxwell 
throughout his work, but \Hdfi. where £f is a factor to be determined. In discussing the steady 
conduction of heat through a gas Clausius assumes that, in addition to the velocity attributed to 
the molecule according to Maxwell's theory, we must also suppose a velocity normal to the stratum 
and depending on the temperature of the stratum. On this assumption the factor H is iuA'estigated 
along with other modifications, and an expression for the assumed velocity is determined from the 
consideration that when the flow of heat is steady there is no movement of the mass. Clausius 
combining his own results with those of Maxwell points out that the expression contained in (28) 
of the paper involves as a result the motion of the gas. He also disputes the accuracy of ex- 
pression (59) for the Conduction of Heat. In the introduction to the memoir published in the 
Phil Trans., 1866, it will be found that Maxwell expresses dissatisfaction with his former theory 
of the Diffusion of Gases, and admits the force of the objections made by Clausius to his expression 
for the Conduction of Heat. Ed.l 



ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GABES. 393 

by the collisions of the opposite stream. When the densities are equal, the 
diffusions due to these two causes respectively are as 2 to 3. 

Prop. XIV. In a system of particles whose density, velocity, &c. are 
functions of x, to find the quantity of matter transferred across the plane of yz, 
due to the motion of agitation alone. 

If the number of particles, their velocity, or their length of path is greater 
on one side of this plane than on the other, then more particles will cross the 
plane in one direction than in the other ; and there will be a transference of 
matter across the plane, the amount of which may be calculated. 

Let there be taken a stratum whose thickness is dx, and 
area unity, at a distance x from the origin. The number of 
collisions taking place in this stratum in unit of time will be 

Njdx. '^^ 

The proportion of these which reach a distance between nl and {n-^dn)l before 

they strike another particle is 

e"" dji. 

The proportion of these which pass through the plane yz is 

nl + x 



2nl 



when X is between —nl and 0, 



and ^r-T- when x is between and + nl ; 

2nl 

the sign being negative in the latter case, because the particles cross the plane 
in the negative direction. The mass of each particle is M ; so that the quantity 
of matter which is projected from the stratum dx, crosses the plane yz in. a. 
positive direction, and strikes other particles at distances between nl and 

(n + dn) I is 

MNvlxTnl) J _„, ,^-s 
2^^ -dxe ""dn (26), 

where x must be between ±nl, and the upper or lower sign is to be taken 
according as x is positive or negative. 

In integrating this expression, we must remember that N, v, and I are 
functions of x, not vanishing with x, and of which the variations are very 
small between the limits x= —nl and x= +nl. 

VOL. L 50 



394 ILLUSTBATIONS OF THE DYNAMICAL THEORY OF GASES. 

As we may have occasion to perform similar integrations, we may state 
here, to save trouble, that if U and r are functions of x not vanishing with x, 
whose variations are very small between the limits x= +r and x= —r, 

/>^^^ = sf2^(^'"") (^^)- 

When m is an odd number, the upper sign only is to be considered; 
when m is even or zero, the upper sign is to be taken with positive values 
of X, and the lower with negative values. Applying this to the case before us, 

We have now to integrate 

n being taken from to oo . We thus find for the quantity of matter trans- 
ferred across unit of area by the motion of agitation in unit of time, 

«=-*s('"'^) (^^)' 

where p = MN is the density, v the mean velocity of agitation, and I the mean 
length of path. 

Prop. XV. The quantity transferred, in consequence of a mean motion of 
translation V, would obviously be 

Q^Vp (29). 

Prop. XVI. To find the resultant dynamical effect of all the collisions 
which take place in a given stratum. 

Suppose the density and velocity of the particles to be functions of x, 
then more particles will be thrown into the given stratum from that side 
on which the density is greatest ; and those particles which have greatest 
velocity will have the greatest effect, so that the stratum will not be generally 



ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES. 395 

in equilibrium, and the dynamical measure of the force exerted on the stratum 
will be the resultant momentum of all the particles which lodge in it during 
unit of time. We shall first take the case in which there is no mean motion 
of translation, and then consider the effect of such motion separately. 

Let a stratum whose thickness is a (a small quantity 
compared with I), and area unity, be taken at the origin, 
perpendicular to the axis of x ; and let another stratum, of 
thickness dx, and area unity, be taken at a distance x from 
the first. 

If M^ be the mass of a particle, N the number in unit of volume, v the 
velocity of agitation,- I the mean length of path, then the number of collisions 
which take place in the stratum dx is 

Njdx, 

The proportion of these which reach a distance between n/ and (n + dn) I is 

e"" dn. 

The proportion of these which have the extremities of their paths in the 
stratum a is 

a 

2nl' 

The velocity of these particles, resolved in the direction of x, is 

vx 
^nl' 

and the mass is M ; so that multiplying all these terms together, we get 

NMv'ax _„ , J /„-.x 

-2^?^' ''^''" <3°> 

for the momentum of the particles fulfilling the above conditions. 

To get the whole momentum, we must first integrate with respect to x 
from x= —nl to x = + nl, remembering that I may be a function of x, and is a 
very small quantity. The result is 

50-2 



396 ILLUSTRATIONS OF THE DYNAMICAL ISHEORY OF GASES. 

Integrating with respect to n from n = to n = co , the result is 

-4A^>^^ ^^^> 

as the -whole resultant force on the stratum a arising from these collisions, 
jyjow =p by Prop. XII., and therefore we may write the equation 



dp 
the ordinary hydrodynamical equation. 



-1=^" (^^)' 



Prop. XVII. To Jind the resultant effect of the collisions upon each of 
several different systems of particles mixed together. 

Let M^, Mj, &c. be the masses of the different kinds of particles, N„ 
N,, &c. the number of each kind in unit of volume, v^, v^, &c. their velocities 
of agitation, Z,, l^ their mean paths, p^, p^, &c. the pressures due to each 
system of particles ; then 



J = Ap^ + Bp^ + &c. 
\=Cp, + Dp, + kc. 



(33). 



The number of collisions of the first kind of particles with each other in unit 
of time will be 

N{OiAp^. 

The number of collisions between particles of the first and second kinds will be 

N{o^Bp^, or N^vJJp^y because v^B=v*C. 

The number of colHsions between particles of the second kind will be 
N^vJ)pi, and so on, if there are more kinds of particles. 

Let us now consider a thin stratum of the mixture whose volume is unity. 

The resultant momentum of the particles of the first kind which lodge in 
it during unit of time is 

dx ' 



ILLU8TRA.TI0NS OF THE DYNAMICAL THEORY OF GASES. 397 

The proportion of these which strike particles of the first kind is 

The whole momentum of these remains among the particles of the first kind. 
The proportion wliich strike particles of the second kind is 

BpA. 

The momentum of these is divided between the striking particles in the ratio 

M 

of their masses ; so that p^ — W of the whole goes to particles of the first 

M 

kind, and -^t^ — ^^, to particles of the second kind. 

Jtf 1 + M, ^ 

The effect of these collisions is therefore to produce a force 

on particles of the first system, and 

on particles of the second system. 

The effect of the collisions of those particles of the second system whic^i 
strike into the stratum, is to produce a force 

on the first system, and 

on the second. 

The whole effect of these collisions is therefore to produce a resultant force 

- 1 (^M.^M ^) - 1 W.^/^c (3.) 

on the first system, 

on the second, and so on. 



398 ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES. 

Prop. XVIII. To find the mechanical effect of a difference in the mean 
velocity of translation of two systems of moving particles. 

Let F,, Fj be the mean velocities of translation of the two systems 

MM 
respectively, then ^ ' ' ( Fj — Fj) is the mean momentum lost by a particle 

of the first, and gained by a particle of the second at collision. The number 
of such collisions in unit of volume is 

NjBp^v,, or N^Cp^v,; 
therefore the whole effect of the collisions is to produce a force 

= -^'^''="-]^^. ('"■-'"•) (*«) 

on the first system, and an equal and opposite force 

= +^=C'p.t..-^^^ (F.- V,) (37) 

on unit of volume of the second system. 

Prop. XIX. To find the law of diffusion in the case of two gases diffu^ng 
into each other through a plug made of a porous material, as in the case of 
the experiments of Graham. 

The pressure on each side of the plug being equal, it was found by Graham 
that the quantities of the gases which passed in opposite directions through the 
plug in the same time were directly as the square roots of their specific gravities. 

We may suppose the action of the porous material to be similar to that 
of a number of particles fixed in space, and obstructing the motion of the 
particles of the moving systems. If Z, is the mean distance a particle of the 
first kind would have to go before striking a fixed particle, and L^ the distance 
for a particle of the second kind, then the mean paths of particles of each 
kind will be given by the equations 

J = ^^, + -Bp, + i, l = Cp, + Z>^, + -i (38). 

The mechanical effect upon the plug of the pressures of the gases on each side, 
and of the percolation of the gases through it, may be found by Props. XVII. 
and XVIII. to be 

M,N,v,V, ^ MJs[,v,V, dp, I dp, k^^ ,3^. 

L, Zj dx Li dx L.i 



ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES. 399 

and this must be zero, if the pressures are equal on each side of the plug. 
Now if Q,, Qj be the quantities transferred through the plug by the mean 
motion of translation, ^, = PiV, = J/jiV, F, ; and since by Graham's law 

we shall have 

M^N{Ui Fi = - MJSf^i\ F, = Z7 suppose ; 

and since the pressures on the two sides are equal, -p= ~~j^> ^^^ ^^® ^^^7 

way in which the equation of equilibrium of the plug can generally subsist is 
when L^ = L^ and l^ = ly This implies that A = C and B = D. Now we know 

that ViB = v*C. Let K=^ —., then we shall have 

A = C=^Kv,\ B = D = ^Kv^ (40), 

and i=i=K{v,p, + i\p,)^-j^ (41). 

The diffusion is due partly to the motion of translation, and partly to that of 
agitation. Let us find the part due to the motion of translation- 

The equation of motion of one of the gases through the plug is found by 
adding the forces due to pressures to those due to resistances, and equating 
these to the moving force, which in the case of slow motions may be neglected 
altogether. The result for jthe first is 



dx 



(^M+^M^^j + fcpA^li,, 



+ ^-^'''*'' -^k (^■- ^=)+ -i-' = '> (*2). 

Making use of the simplifications we have just discovered, this becomes 

^ ^^ {v,% + v:p:) + K -^, (p,v, +p,v,) U + yU (43), 

whence l^= -^ ia(v,^p,^v,%) 

A^iVj {p^V^ +i?aVi) + f~ 



400 ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES. 

whence the rate of diffusion due to the motion of translation may be found ; for 

(?. = J, andft=-J (45). 

To find the difiusion due to the motion of agitation, we must find the 
value of q^. 

L d p. 



V, dx 1+ KL (v,p^ + v^p,) ' 

^'--.t1I^i+^^^(^^-^^^» ('')• 

SimHarly, q,= + l^^{l+KLi^{p,+p:)} (47). 

The whole diffusions are Q^ + q, and Q, + q,. The values of q, and q, have a 

term not following Graham's law of the square roots of the specific gravities, 

but following the law of equal volumes. The closer the material of the plug, 
the less will this term affect the result. 

Our assumptions that the porous plug acts like a system of fixed particles, 
and that Graham's law is fulfilled more accurately the more compact the 
material of the plug, are scarcely sufficiently well verified for the foundation of 
a theory of gases ^ and even if we admit the original assumption that they are 
systems of moving elastic particles, we have not very good evidence as yet for 
the relation among the quantities A, B, C, and D. 

Prop. XX. To find the rate of diffusion between two vessels connected hy a 
tube. 

When diffusion takes place through a large opening, such as a tube con- 
necting two vessels, the question is simplified by the absence- of the porous 
diffusion plug; and since the pressure is constant throughout the apparatus, the 
volumes of the two gases passing opposite ways through the tube at the same 
time must be equal Now the quantity of gas which passes through the tube 
is due partly to the motion of agitation as in Prop. XIV., and partly to the 
mean motion of translation as in Prop. XV. 



ILLUSTRATIONS OF THE DYNAMICAL THEORY OF OASES. 401 

Let US suppose the volumes of the two vessels to be a and h, and the 
length of the tube between them c, and its trans- 
verse section s. Let a be filled with the first gas, /^ * ^ /^ 
and h with the second at the commencement of 
the experiment, and let the pressure throughout 
the apparatus be P. 

Let a volume y of the first gas pass from a to 6, and a volume y of the 
second pass from h to a \ then if p, and p^ represent the pressures in a. due 
to the first and second kinds of gas, and p\ and p\ the same in the vessel h, 




r>='±^:yp r)=y-P r>'=y-P V'^—^P {i%\ 



Since there is still equilibrium, 

which gives y = y and p^ +^, = P =p\ ■\-p„ (49). 

The rate of diffusion will be +-^ for the one gas, and —-— for the other, 
measured in volume of gas at pressure P. 

Now the rate of diflfusion of the first gas will be 

dji_^iji,±pj,_^-±yp^'^^^^ 

dt~' p -' — p — (50)' 

and that of the second, 

-di=' p (='i)- 

We have also the equation, derived from Props. XVI. and XVIL, 

^ {Ap,l, (M, + if,) + BplM, - CpJ^M} + Bp,p,vM{ F. - F,) = (52). 

From these three equations we can eliminate F, and V., and find -^ in 

ift 



terms of p and -j- , so that we may w^rite 



S=/(^"S) (-)• 



VOL. I. 51 



402 ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES. 

Since the capacity of the tube is small compared with that of the vessels, 

we may consider -^ constant through the whole length of the tube. "We may 

then solve the differential equation in p and x; and then making p=Pi when 
x = 0, and p=Pi when x = c, and substituting for p^ and p\ their values in 
terms of y, we shall have a differential equation in y and t, which being solved, 
will give the amount of gas diffused in a given time. 

The solution of these equations would be difficult unless we assume rela- 
tions among the quantities Ay B, C, D, which are not yet sufficiently estab- 
lished in the case of gases of different density. Let us suppose that in a 
particular case the two gases have the same density, and that the four quan- 
tities A, B, Cy D are all equal. 

The volume diffused, owing to the motion of agitation of the particles, is 
then 

3 P dx '''' 
and that due to the motion of translation, or the interpenetration of the two 

gases in opposite streams, is 

5 dp kl 
P dx V ' 

The values of v are distributed according to the law of Prop. IV., so that 

the mean value oi v is -i^ , and that of - is -7=- , that of k being \a^. The 
VTT V Vira 

diffusions due to these two causes are therefore in the ratio of 2 to 3, and 

their sum is 

dy _ ^ J2k si dp , . 

di-~^s]~^Pdx ^^^^• 

If we suppose -^ constant throughout the tube, or, in other words, if we 
regard the motion as steady for a short time, then -r- will be constant and 
equal to — — —\ or substituting from (48), 



ah ,, ~t^ 



(a+6)^ 



whence y = — /(I— e"" "*** ) (56). 



ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES. 403 

By choosing pairs of gases of equal density, and ascertaining the amount 
of diffusion in a given time, we might determine the value of I in this expres- 
sion. The diffusion of nitrogen into carbonic oxide or of deutoxide of nitrogen 
into carbonic acid, would be suitable cases for experiment. The only existing 
experiment which approximately fulfils the conditions is one by Graham, quoted 
by Herapath from Brande's Quarterly Journal of Science, Vol. xviiL p. 7Q. 

A tube 9 inches long and 0*9 inch diameter, communicated with the 
atmosphere by a tube 2 inches long and 0'12 inch diameter; 152 parts of 
olefiant gas being placed in the tube, the quantity remaining after four hours 
was 9 9. parts. 

In this case there is not much difference of specific gravity between the 

and we have a = 9 x (0'9)'' - cubic inches, 2^=00, c = 2 inches, and 



(0*12)' - square inches; 






^^ log. 10.^. log.. (^^) (57); 

.-. ^ = 0-00000256 inch =39^000 i"ch (58). 



Prop. XXI. To Jind the amount of energy which crosses unit of area in 
unit of time when the velocity of agitation is greater on one side of the area 
than on the other. 

The energy of a single particle is composed of two parts, — the vis viva 
of the centre of gravity, and the vis viva of the various motions of rotation 
round that centre, or, if the particle be capable of internal motions, the vis 
viva of these. We shall suppose that the whole vis viva bears a constant 
proportion to that due to the motion of the centre of gravity, or 

where )8 is a coefficient, the experimental value of which is 1*634. Substituting 
E for Ji" in Prop. XIV., we get for the transference of energy across unit 
of area in unit of time, 

51—2 



404 ILLUSTRATIONS OF THE DYNAMICAI, THEORY OF GASES. 

where J is the mechanical equivalent of heat in foot-pounds, and q[ is the 
transfer of heat in thermal units. 

Now MN=p, and l = -i-, so that MNl = -. ; 
'^^ Ap A 

••••^^=-*'^l (-)■ 

Also, if T is the absolute temperature, 

1 dT^2dv^^ 
T dx~ V dx' 

.■.Jq= -ify.lv ^"^ (60), 

where p must be measured in dynamical units of force. 

Let J =772 foot-pounds, _p = 2116 pounds to square foot, ^ = 4:ooVoo i^^^^' 
v=1505 feet per second, T=522 or 62" Fahrenheit; then 

2=;« (">' 

where q is the flow of heat in thermal units per square foot of area ; and T' 
and T are the temperatures at the two sides of a stratum of air x inches thick. 

In Prof. Rankine's work on the Steam-engine, p. 259, values of the thennal 
resistance, or the reciprocal of the conductivity, are given for various substances 
as computed from a Table of conductivities deduced by M. Peclet from experi- 
ments by M. Despretz : — 

Resistance. 

Gold, Platinum, Silver 0-0036 

Copper 0-0040 

Iron 0-0096 

Lead 0-0198 

Brick 0-3306 

Ail' by our calculation 40000 

It appears, therefore, that the resistance of a stratum of air to the con- 
duction of heat is about 10,000,000 times greater than that of a stratum of 



ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES. 405 

copper of equal thickness. It would be almost impossible to establish the value 
of the conductivity of a gas by direct experiment, as the heat radiated from the 
sides of the vessel would be far greater than the heat conducted through the 
air, even if currents could be entirely prevented*. 



PART III. 

ON THE COLLISION OF PERFECTLY ELASTIC BODIES OF ANY FORM. 

When two perfectly smooth spheres strike each other, the force which acts 
between them always passes through their centres of gravity ; and therefore their 
motions of rotation, if they have any, are not affected by the collision, and 
do not enter into our calculations. But, when the bodies are not spherical, 
the force of compact will not, in general, be in the line joining their centres 
of gravity ; and therefore the force of impact will depend both on the motion 
of the centres and the motions of rotation before impact, and it will affect 
both these motions after impact. . 

In this way the velocities of the centres and the velocities of rotation 
will act and react on each other, so that finally there will be some relation 
established between them ; and since the rotations of the particles about their 
three axes are quantities related to each other in the same way as the three 
velocities of their centres, the reasoning of Prop. IV. will apply to rotation as 
well as velocity, and both will be distributed according to the law 

dN ^r 1 -- 
-T- = i V — j^ e *' . 
ax a. 'Ju 

* [Clausius, in the memoir cited in the last foot-note, has pointed out two oversights in this 
calculation. In the first place the numbers have not been proi^erly reduced to English measure, 
and have still to be multiplied by 4356, the ratio of the English pound to the kilogramme. The 
numbers have, further, been calculated with one hour as the unit of time, whereas Maxwell h>\s 
used them as if a second had been the unit. Taking account of these circumstarces and using his 
own expression for the conduction which differs from (59) only in haNnng ^V in place of ^ on the 
right-hand side, Clausius finds that the resistance of a stratum of air to the conduction of heat is 
1400 times greater than that of a stratum of lead of the same thickness, or about 7000 times greater 
than that of copper. Ed.] 



406 ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES. 

Also, by Prop. V., if a; be tbe average velocity of one set of particles, and y 
that of another, then the average value of the sum or difference of the velocities is 

from which it is easy to see that, if in each individual case 

w = ax + fey + cz, 

where x, y, z are independent quantities distributed according to the law above 
stated, then the average values of these quantities will be connected by the 
equation 

Prop. XXII. Two perfectly elastic bodies of any form strike each other: 
given their motions before impact, and the line of i^npact, to find their motions 
after impact. 

Let M, and M, be the centres of gravity of the two bodies. M,X„ M,Y„ 
and i¥jZ, the principal axes of the first; and MJC^, 
M,Y, and M^, those of the second. Let / be the 
point of impact, and EJE, the line of impact. 

Let the co-ordinates of / with respect to if, be 
x^,z„ and with respect to M^ let them be x.^.jt,. 

Let the direction-cosines of the line of impact 
RJR, be l,m,n, with respect to M„ and l,7n,n, with 
respect to M^. 

^ Let M, and M, be the masses, and A.B^ and A,BA the moments of 
inertia of the bodies about their principal axes. 

Let the velocities of the centres of gravity, resolved in the direction of 
the principal axes of each body, be 

Z7„ F„ W„ and U,, V„ Tr„ before impact, 
^^^ ^» y» W\, and ir„ F„ W'„ after impact. 

Let the angular velocities round the same axes be 

Pi, q^ r„ and p„ q„ r„ before impact, 
^^^ P\> ?'i. f^'i, and p\, q\, r^ after impact. 




ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES. 407 

Let R be the impulsive force between the bodies, measured by the momentum 
it produces in each. 

Then, for the velocities of the centres of gravity, we have the following 
equations : 

^'■= ^'+f'' ^'•= ^'-K (^2), 

with two other pairs of equations in V and W. 
The equations for the angular velocities are 

p\ =Pi + -J (y^n, - z,m,), p, =p, - -J (y,n, - z,m,) (63), 

with two other pairs of equations for q and r. 

The condition of perfect elasticity is that the whole vis viva shall be the 
same after impact as before, which gives the equation 

M, ( U\ - U\) + M, ( U'\ - U\) + A, {p\ -p\) + A, {p\ -p\) + &c. = 0. . . . (64). 

The terms relating to the axis of x are here given ; those relating to y and 
z may be easily written down. 

Substituting the values of these terms, as given by equations (62) and (63), 
and dividing by R, we find 

h{U\+ U,)-k{U\+ U,) + (y,n,-z,m,)(p\+p,)-{y,n,-z,m,) (p\+p,) + &c. = 0...{e5). 

Now if v^ be the velocity of the striking-point of the first body before 
impact, resolved along the line of impact, 

v^ = lJJ^-\- (y^Tii — z^mi) pi + &c. ; 

and if we put v^ for the velocity of the other striking-point resolved along the 
same line, and v\ and v\ the same quantities after impact, we may write, 
equation (65), 

v^-\-v\ — v^ — v\ = (66), 

or v^-Vj = v\-v\ (67), 

which shows that the velocity of separation of the striking-points resolved in 
the line of impact is equal to that of approach. 



408 ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES. 

Substituting the values of the accented quantities in equation (65) by means 
of equations (63) and (64), and transposing terms in J?, we find 
2 {UJ, - UJ, +Pi {y,n, - z,m,) -p, {y,n, - zjn,)} 4- &c. 

the other terms being related to y and z as these are to x. From this equation 
we may find the value of E ; and by substituting this in equations (63), (64), 
we may obtain the values of all the velocities after impact. 

"We may, for example, find the value of U\ from the equation 

ir (^' , 4' , {y.n,-z,m,Y . {y.n.-z.'m^Y ] M, ] 

^^\M^M^ A, + A, ^^7T 

-^a M^M^ — A — ^ — A — "^^'TT 

+ 2 U,l, - 2p, {y,n, - z,m,) + 2p, (y^i, - z,m,) - &c. 



(69). 



Prop. XXIII. To find the relations between the average velocities of trans- 
lation and rotation after many collisions among many bodies. 

Taking equation (69), which applies to an individual collision, we see that 
U\ is expressed as a linear function of Z7„ U„ p„ p„ &c., all of which are 
quantities of which the values are distributed among the different particles 
according to the law of Prop. IV. It follows from Prop. V., that if we square 
every term of the equation, we shall have a new equation between the average 
values of the different quantities. It is plain that, as soon as the required 
relations have been estabUshed, they will remain the same after collision, so that 
we may put Z7;"= U,' in the equation of averages. The equation between the 
average values may then be written 

Now since there are collisions in every possible way, so that the values of 
I, m, n, &c. and x, y, z, &c. are infinitely varied, this equation cannot subsist 
unless 

The final state, therefore, of any number of systems of moving particles of 
any form is that in which the average vis viva of translation along each of the 



ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES. 409 

three axes ia the same in all the systems, and equal to the average vis viva 
of rotation about each of the three principal axes of each particle. 

Adding the tires vivcn with respect to the other axes, we find that the 
whole via viva of translation is equal to that of rotation in each system of 
particles, and is also the same for different systems, as was proved in Prop. VI. 

This result (which is true, however nearly the bodies approach the spherical 
form, provided the motion of rotation is at all afiected by the collisions) seems 
decisive against the unqualified acceptation of the hypothesis that gases are such 
systems of hard elastic particles. For the ascertained fact that y, the ratio of 
the specific heat at constant pressure to that at constant volume, is equal to 
1-408, requires that the ratio of the whole vis viva to the vis viva of translation 
should be 

^ = 3(^) = '-«^^^ 
whereas, according to our hypothesis, )S = 2 . 

We have now followed the mathematical theory of the collisions of hard 
elastic particles through various cases, in which there seems to be an analogy 
with the phenomena of gases. We have deduced, as others have done already, 
the relations of pressure, temperature, and density of a single gas. We have 
also proved that when two different gases act freely on each other (that is, when 
at the same temperature), the mass of the single particles of each is inversely 
proportional to the square of the molecular velocity ; and therefore, at equal 
temperature and pressure, the number of particles in unit of volume is the same. 

We then offered an explanation of the internal friction of gases, and deduced 
from experiments a value of the mean length of path of a particle between 
successive collisions. 

We have applied the theory to the law of diffusion of gases, and, from an 
experiment on olefiant gas, we have deduced a value of the length of path not 
very different from that deduced from experiments on friction. 

Using this value of the length of path between collisions, we found that the 
resistance of air to the conduction of heat is 10,000,000 times that of copper, a 
result in accordance with experience. 

Finally, by establishing a necessary relation between the motions of trans- 
lation and rotation of all particles not spherical, we proved that a system of 
such particles could not possibly satisfy the known relation between the two 
specific heats of all gases. 



[From the Philosophical Transactions, MDCCCLX.] 



XXI. On the Theory of Compound Colours, and the Relations of the Colours 
of the Spectrum. Communicated by Professor Stokes, Sec. R.S. 

(Received January 5,— Read March 22, 1860.) 



§ I. Introduction. 

According to Newton's analysis of light ^', every colour in nature is pro- 
duced by the mixture, in various proportions, of the different kinds of light 
into which white %ht is divided by refraction. By means of a prism we may 
analyse any coloured light, and determine the proportions in which the different 
homogeneous rays enter into it; and by means of a lens we may recombine 
these rays, and reproduce the original coloured light. 

Newton has also shewnt how to combine the different rays of the spectrum 
80 as to form a single beam of light, and how to alter the proportions of the 
different colours so as to exhibit the result of combining them in any arbitrary 
manner. 

The number of different kinds of homogeneous light being infinite, and the 
proportion in which each may be combined being also variable indefinitely, the 
results of such combinations could not be appreciated by the eye, unless the 
chromatic effect of every mixture, however complicated, could be expressed in 
some simpler form. Colours, as seen by the human eye of the normal type, can 
all be reduced to a few classes, and expressed by a few well-known names; and 
even those colours which have different names have obvious relations among them- 
selves. Every colour, except purple, is similar to some colour of the spectrum |, 

* Optics, Book I. Part 2, Prop. 7. 

t Lectiones Opticce, Part 2, § 1, pp. 100 to 105; and Optics, Book i. Part 2, Prop. 11. 

X Optica, Book L Part 2, Prop. 4. 



ON THE THEORY OF COMPOUND COLOURS. 411 

although less intense ; and all purples may be compounded of blue and red, 
and diluted with white to any required tint. Brown colours, which at first 
sight seem different, are merely red, orange or yellow of feeble intensity, more 
or less diluted with white. 

It appears therefore that the result of any mixture of colours, however 
complicated, may be defined by its relation to a certain small number of 
well-known colours. Having selected our standard colours, and determined the 
relations of a given colour to these, we have defined that colour completely as 
to its appearance. Any colour which has the same relation to the standard 
colours, will be identical in appearance, though its optical constitution, as 
revealed by the prism, may be very different. 

We may express this by saying that two compound colours may be chro- 
matically identical, but optically different. The optical properties of light are 
those which have reference to its origin and propagation through media, till it 
falls on the sensitive organ of vision; the chromatical properties of light are 
those which have reference to its power of exciting certain sensations of colour, 
perceived through the organ of vision. 

The investigation of the chromatic relations of the rays of the spectrum 
must therefore be founded upon observations of the apparent identity of com- 
pound colours, as seen by an eye either of the normal or of some abnormal 
type; and the results to which the investigation leads must be regarded as 
partaking of a physiological, as well as of a physical character, and as indicating 
certain laws of sensation, depending on the constitution of the organ of vision, 
which may be different in different individuals. We have to determine the 
laws of the composition of colours in general, to reduce the number of standard 
colours to the smallest possible, to discover, if we can, what they are, and to 
ascertain the relation which the homogeneous light of different parts of the 
spectrum bears to the standard colours. 

§ II. History of the Theory of Compound Colours. 

The foundation of the theory of the composition of colours was laid by 
Newton*. He first shews that, by the mixture of homogeneal light, colours 
may be produced which are "like to the colours of homogeneal light as to 
the appearance of colour, but not as to the immutabOity of colour and consti- 

* Optics, Book I. Part 2, Props. 4, 5, 6. 



412 ON THE THEORY OF COMPOUND COLOURS. 

tution of light." Red and yellow give an orange colour, which is chromatically 
similar to the orange of the spectrum, but optically different, because it is 
resolved into its component colours by a prism, while the orange of the spectrum 
remains unchanged. When the colours to be mixed lie at a distance from one 
another in the spectrum, the resultant appears paler than that intermediate 
colour of the spectrum which it most resembles; and when several are mixed, 
the resultant may appear white. Newton* is always careful, however, not to 
call any mixture white, unless it agrees with comnon white light in its optical 
as well as its chromatical properties, and is a mixture of all the homogeneal 
colours. The theory of compound colours is first presented in a mathematical 
form in Prop. 6, " In a mixture of priinary colours, the quantity arid quality 
of each being given, to know the colour of the compound." He divides the 
circumference of a circle into seven parts, proportional to the seven musical 
intervals, in accordance with his opinion about the proportions of the colours 
in the spectrum. At the centre of gravity of each of these arcs he places a 
little circle, whose area is proportional to the number of rays of the corre- 
sponding colour which enter into the given mixture. The position of the centre 
of gravity of all these circles indicates the nature of the resultant colour. A 
radius drawn through it points out that colour of the spectrum which it most 
resembles, and the distance from the centre determines the fulness of its colour. 

With respect to this construction, Newton says, " This rule I conceive 
accurate enough for practice, though not mathematically accurate." He gives no 
reasons for the different parts of his rule, but we shall find that his method 
of finding the centre of gravity of the component colours is completely con- 
firmed by my observations, and that it involves mathematically the theory of three 
elements of colour ; but that the disposition of the colours on the circumference 
of a circle was only a provisional arrangement, and that the true relations of 
the colours of the spectrum can only be determined by direct observation. 

Young t appears to have originated the theory, that the three elements of 
colour are determined as much by the constitution of the sense of sight as by 
anything external to us. He conceives that three different sensations may be 
excited by light, but that the proportion in which each of the three is excited 
depends on the nature of the light. He conjectures that these primary sensa- 

* 7th and 8th Letters to Oldenburg. 

+ Young's Lectures on Natural Philosophy, Kelland's Edition, p. 345, or Quarto, 1807, Vol. i. 
p. 441 ; see also Young in Philosophical Transaction, 1801, or Works in Quarto, Vol il. p. 617. 



ON THE THEORY OF COMPOUND COLOURS. 413 

tions correspond to red, green, and violet. A blue ray, for example, though 
homogeneous in itself, he conceives capable of exciting both the green and the 
violet sensation, and therefore he would call blue a compound colour, though 
the colour of a simple kind of light. The quality of any colour depends, 
according to this theory, on the ratios of the intensities of the three sensations 
which it excites, and its bHghtness depends on the sum of these three intensities. 
Sir David Brewster, in his paper entitled " On a New Analysis of Solar 
Light, indicating three Primary Colours, forming Coincident Spectra of equal 
length*," regards the actual colours of the spectrum as arising from the inter- 
mixture, in various proportions, of three primary kinds of light, red, yellow, 
and blue, each of which is variable in intensity, but uniform in colour, from 
one end of the spectrum to the other ; so that every colour in the spectrum 
is really compound, and might be shewn to be so if we had the means of 
separating its elements. 

Sir David Brewster, in his researches, employed coloured media, which, 
according to him, absorb the three elements of a single prismatic colour in 
different degrees, and change their proportions, so as to alter the colour of the 
light, without altering its refrangibility. 

In this paper I shall not enter into the very important questions affecting 
the physical theory of light, which can only be settled by a careful inquiry 
into the phenomena of absorption. The physiological facts, that we have a 
threefold sensation of colour, and that the three elements of this sensation are 
affected in different proportions by light of different refrangibilities, are equally 
true, whether we adopt the physical theory that there are three kinds of light 
corresponding to these three colour-sensations, or whether we regard light of 
definite refrangibility as an undulation of known length, and therefore variable 
only in intensity, but capable of producing difierent chemical actions on different 
substances, of being absorbed in different degrees by different media, and of 
exciting in different degrees the three different colour-sensations of the human 
eye. 

Sir David Brewster has given a diagram of three curves, in which the 
base-line represents the length of the spectrum, and the ordinates of the curves 
represent, by estimation, the intensities of the three kinds of light at each point 
of the spectrum. I have employed a diagram of the same kind to express the 

* Transactions of the Royal Society of Edivimrgh, Vol. xii. p. 123. 



414 ON THE THEORY OF COMPOUND COLOURS. 

results arrived at in this paper, the ordinates being made to represent the 
intensities of each of the three elements of colour, as calculated from the 
experiments. 

The most complete series of experiments on the mixture of the colours of 
the spectrum, is that of Professor Helmholtz*, of Konlgsberg. By using two 
sHts at right angles to one another, he formed two pure spectra, the fixed 
lines of which were seen crossing one another when viewed in the ordinary- 
way by means of a telescope. The colours of these spectra were thus combined 
in every possible way, and the effect of the combination of any two could be 
seen separately by drawing the eye back from the eye-piece of the telescope^ 
when the compound colour was seen by itself at the eye-hole. The proportion 
of the components was altered by turning the combined slits round in their 
own plane. 

One result of these experiments was, that a colour, chromatically identical 
with white, could be formed by combining yellow with indigo. M. Helmholtz. 
was not then able to produce white with any other pair of simple colours, and 
considered that three simple colours were required in general to produce white^ 
one from each of the three portions into which the spectrum is divided by 
the yellow and indigo. 

Professor Grassmannf shewed that Newton's theory of compound colours 
implies that there are an infinite number of pairs of complementary colours in 
the spectrum, and pointed out the means of finding them. He also shewed 
how colours may be represented by lines, and combined by the method of the 
parallelogram. 

In a second memoirj, M. Helmholtz describes his method of ascertaining 
these pairs of complementary colours. He formed a pure spectrum by means 
of a slit, a prism, and a lens ; and in this spectrum he placed an apparatus 
having two parallel slits which were capable of adjustment both in position 
and breadth, so as to let through any two portions of the spectrum, in any 
proportions. Behind this slit, these rays were united in an image of the prism, 
which was received on paper. By arranging the slits, the colour of this image 
may be reduced to white, and made identical with that of paper illuminated with 
white light. The wave-lengths of the component colours were then measured by 
observing the angle of diffraction through a grating. It was found that the 

* Poggendorffs Anncden, Band lxxxvil {Philosophical Magazine, 1852, December). 

t Ibid. Band lxxxix. (Philosophical Magazine, 1854, April). J Ibid. Band xciv. 



ON THE THEORY OF COMPOUND COLOURS. 415 

colours from red to green-yellow (X=2082) were complementary to colours ranging 
from green-blue (X=1818) to violet, and that the colours between green-yellow 
and green-blue have no homogeneous complementaries, but must be neutralized 
by mixtures of red and violet. 

M. Helmholtz also gives a provisional diagram of the curve formed by the 
spectrum on Newton's diagram, for which his experiments did not furnish him 
with the complete data. 

Accounts of experiments by myself on the mixture of artificial colours by 
rapid rotation, may be found in the Transactions of the Royal Society of 
Edinburgh, Vol. xxi. Pt. 2 (1855); in an appendix to Professor George Wilson's 
work on Coloiu--Blindness ; in the Report of the British Association for 1856, 
p. 12; and in the Philosophical Magazine, July 1857, p. 40. These experiments 
shew that, for the normal eye, there are three, and only three, elements of 
colour, and that in the colour-blind one of these is absent. They also prove 
that chromatic observations may be made, both by normal and abnormal eyes, 
with such accuracy, as to warrant the employment of the results in the calcu- 
lation of colour-equations, and in laying down colour-diagrams by Newton's rule. 

The first instrument which I made (in 1852) to examine the mixtures of 
the colours of the spectrum was similar to that which I now use, but smaller, 
and it had no constant light for a term of comparison. The second was 6^ feet 
long, made in 1855, and shewed tico combinations of colour side by side. I have 
now succeeded in making the mixture much more perfect, and the comparisons 
more exact, by using white reflected light, instead of the second compound 
colour. An apparatus in which the light passes through the prisms, and is 
reflected back again in nearly the same path by a concave mirror, was shewn 
by me to the British Association in 1856. It has the advantage of being 
portable, and need not be more than half the length of the other, in order 
to produce a spectrum of equal length. I am so well satisfied with the working 
of this form of the instrument, that I intend to make use of it in obtaining 
equations from a gi-eater variety of observers than I could meet with when I 
was obliged to use the more bulky instrument. It is difficult at first to get 
the observer to believe that the compound light can ever be so adjusted as to 
appear to his eyes identical with the white light in contact with it. He has to 
learn what adjustments are necessary to produce the requisite alteration under 
all circumstances, and he must never be satisfied till the two parts of the 
field are identical in colour and illumination. To do this thoroughly, implies 



416 ON THE THEORY OF COMPOUND COLOURS. 

not merely good eyes, but a power of judging as to the exact nature of the 
difference between two very pale and nearly identical tints, whether they differ 
in the amount of red, green, or blue, or in brightness of illumination. 

In the following paper I shall first lay down the mathematical theory of 
Newton's diagram, with its relation to Young's theory of the colour-sensation. 
I shall then describe the experimental method of mixing the colours of the 
spectrum, and determining the wave-lengths of the colours mixed. The results 
of my experiments will then be given, and the chromatic relations of the 
spectrum exhibited in a system of colour-equations, in Newton's diagram, and 
in three curves of intensity, as in Brewster's diagram. The differences between 
the results of two observers will then be discussed, shewing on what they 
depend, and in what way such differences may affect the vision of persons 
othei-wise free from defects of sight. 



§ III. Mathematical Theory of Newton's Diagram of Colours. 

Newton's diagram is a plane figure, designed to exhibit the relations of 
colours to each other. 

Every point in the diagram represents a colour, simple or compound, and 
we may conceive the diagram itself so painted, that every colour is found at 
its corresponding point. Any colour, differing only in quantity of illumination 
from one of the colours of the diagram, is referred to it as a unit, and is 
measured by the ratio of the illumination of the given colour to that of the 
corresponding colour in the diagram. In this way the quantity of a colour is 
estimated. The resultant of mixing any two colours of the diagram is found 
by dividing the line joining them inversely as the quantity of each; then, if 
the sum of these quantities is unity, the resultant will have the illumination 
as weU as the colour of the point so found; but if the sum of the components 
is different from unity, the quantity of the resultant will be measured by the 
sum of the components. 

This method of determining the position of the resultant colour is mathe- 
matically identical with that of finding the centre of gravity of two weights, 
and placing a weight equal to their sum at the point so found. We shall 
therefore speak of the resultant tint as the sum of its components placed at 
their centre of gravity. 



ON THE THEORY OF COMPOUND COLOURS. 41/ 

By compounding this resultant tint with some other colour, we may find the 
position of a mixture of three colours, at the centre of gravity of its components ; 
and by taking these components in different proportions, we may obtain colours 
corresponding to every part of the triangle of which they are the angular points. 
In this way, by taking any three colours we should be able to construct a 
triangular portion of Newton's diagram by painting it with mixtures of the three 
colours. Of course these mixtures must be made to correspond with optical 
mixtures of light, not with mechanical mixtures of pigments. 

Let us now take any colour belonging to a point of the diagram outside 
this triangle. To make the centre of gravity of the three weights coincide with 
this point, one or more of the weights must be made negative. This, though 
following from mathematical principles, is not capable of direct physical inter- 
pretation, as we cannot exhibit a negative colour. 

The equation between the three selected colours, x, y, z, and the new colour 
u, may in the first case be written 

u = x + y-\-z (1), 

05, y, % being the quantities of colour required to produce u. In the second case 
suppose that z must be made negative, 

u = x-^-y — z (2). 

As we cannot realize the term — z as a negative colour, we transpose it to the 
other side of the equation, which then becomes 

u-\-z = x-\-y (3), 

which may be interpreted to mean, that the resultant tint, u + z, is identical 
with the resultant, x-\-y. We thus find a mixture of the new colour with one 
of the selected colours, which is chromatically equivalent to a mixture of the 
other two selected colours. 

When the equation takes the form 

u — x — y — z (4), 

two of the components being negative, we must transpose them thus, 

u + y-\-z = x (5), 

which means that a mixture of certain proportions of the new colour and two 
of the three selected, is chromatically equivalent to the third. We may thus in 
all cases find the relation between any three colours and a fourth, and exhibit 



418 ON THE THEORY OF COMPOUND COLOURS. 

this relation in a form capable of experimental verification; and by proceeding 
in this way we may map out the positions of all colours upon Newton's diagram. 
Every colour in nature will then be defined by the position of the corresponding 
colour in the diagram, and by the ratio of its illumination to that of the 
colour in the diagram. 



§ lY. Method of representing Colours by Straight Lines drawn from a Point. 

To extend our ideas of the relations of colours, let us form a new geome- 
trical conception by the aid of solid geometry. 

Let us take as origin any point not in the plane of the diagram, and let 
us draw lines through this point to the different points of the diagram; then 
the direction of any of these lines will depend upon the position of the point 
of the diagram through which it passes, so that we may take this line as the 
representative of the corresponding colour on the diagram. 

In order to indicate the quantity of this colour, let it be produced beyond 
the plane of the diagram in the same ratio as the given colour exceeds in 
illumination the colour on the diagram. In this way every colour in nature will 
be represented by a line drawn through the origin, whose direction indicates 
the quality of the colour, while its length indicates its quantity. 

Let us find the resultant of two colours by this 
method Let O be the origin and AB be a section 
of the plane of the diagram by that of the paper. 
Let OP, 0^ be lines representing colours, A, B the 

OP 

corresponding points in the diagram ; then the quantity of P will be jr-^ —P> 

and that of Q will be jyD = 9.- The resultant of these will be represented in 

the diagram by the point C, where AC : CB wq-.p, and the quantity of the 
resultant will be p + q, so that if we produce OC to R, so that OR = (p-\-q)OC, 
the line OR will represent the resultant of OP and OQ in direction and 
magnitude. It is easy to prove, from this construction, that OR is the diagonal 
of the parallelogram of which OP and OQ are two sides. It appears therefore 
that if colours are represented in quantity and quality by the magnitude and 
direction of straight lines, the rule for the composition of colours is identical 




ON THE THEORY OF COMPOUND COLOURS. 419 

witli that for the composition of forces in mechanics. This analogy has been 
well brought out by Professor Grassmann in Poggendorflf's Annalen, Bd. lxxxix. 

We may conceive an arrangement of actual colours in space founded upon 
this construction. Suppose each of these radiating lines representing a given 
colour to be itself illuminated with that colour, the brightness increasing from 
zero at the origin to unity, where it cuts the plane of the diagram, and 
becoming continually more intense in proportion to the distance from the origin. 
In this way every colour in nature may be matched, both in quaUty and 
quantity, by some point in this coloured space. 

If we take any three lines through the origin as axes, we may, by co-ordi- 
nates parallel to these lines, express the position of any point in space. That 
point will correspond to a colour which is the resultant of the three colours 
represented by the three co-ordinates. 

This system of co-ordinates is an illustration of the resolution of a colour 
into three components. According to the theory of Young, the human eye is 
capable of three distinct primitive sensations of colour, which by their composition 
in various proportions, produce the sensations of actual colour in all their varieties. 
Whether any kinds of light have the power of exciting these primitive sensations 
separately, has not yet been determiaed. 

If colours corresponding to the three primitive sensations can be exhibited, 
then all colours, whether produced by light, disease, or imagination, are com- 
pounded of these, and have their places within the triangle formed by joining 
the three primaries. If the colours of the pure spectrum, as laid down on the 
diagram, form a triangle, the colours at the angles may correspond to the primitive 
sensations. K the curve of the spectrum does not reach the angles of the circum- 
scribing triangle, then no coloiir in the spectrum, and therefore no colour in 
nature, corresponds to any of the three primary sensations. 

The only data at present existing for determining the primary colours, are 
derived from the comparison of observations of colour-equations by colour-blind, 
and by normal eyes. The colour-blind equations ditfer from the others by the 
non-existence of one of the elements of colour, the relation of which to known 
colours can be ascertained. It appears, from observations made for me by two 
colour-blind persons*, that the elementary sensation which they do not possess 
is a red approaching to crimson, lying beyond both vermilion and carmine. These 

♦ Trfmsactiona of the Royal Society of Edinburgh, Vol. xiL Pt 2, p. 286. 



420 ON THE THEORY OF COMPOUND COLOURS. 

observations are confirmed by those of Mr Pole, and by others which I have 
obtained since. I have hopes of being able to procure a set of colour-blind 
equations between the colours of the spectrum, which will indicate the missing 
primary in a more exact manner. 

The experiments which I am going to describe have for their object the 
determination of the position of the colours of the spectrum upon Newton's 
diagram, from actual observations of the mixtures of those colours. They were 
conducted in such a way, that in every observation the judgment of the observer 
was exercised upon two parts of an illuminated field, one of which was so 
adjusted as to be chromatically identical with the other, which, during the whole 
series of observations, remained of one constant intensity of white. In this way 
the efiects of subjective colours were entirely got rid of, and all the observa- 
tions were of the same kind, and therefore may claim to be equally accurate ; 
which is not the case when comparisons are made between bright colours of 
different kinds. 

The chart of the spectrum, deduced from these observations, exhibits the 
colours arranged very exactly along two sides of a triangle, the extreme red and 
violet forming doubtful portions of the third side. This result greatly simplifies 
the theory of colour, if it does not actually point out the three primary colours 
themselves. 



§ V. Description of an Instruinent for making definite Mixtures of the 
Colours of the Spectrum. 

The experimental method which I have used consists in forming a combi- 
nation of three colours belonging to different portions of the spectrum, the quantity 
of each being so adjusted that the mixture shall be white, and equal in intensity 
to a given white. Fig. 1, Plate VI. p. 444, represents the instrument for 
making the observations. It consists of two tubes, or long boxes, of deal, of 
rectangular section, joined together at an angle of about 100". 

The part AK is about five feet long, seven inches broad, and four deep ; 
KN is about two feet long, five inches broad, and four deep ; BD is a partition 
parallel to the side of the long box. The whole of the inside of the instrument 
is painted black, and the only openings are at the end AC, and at E. At the 
angle there is a Hd, which is opened when the optical parts have to be adjusted 
or cleaned. 



ON THE THEORY OF COMPOUND COLOURS. 421 

At -£^ is a fine vertical slit ; Z is a lens ; at P there are two equilateral 
prisms. The slit E, the lens L, and the prisms P are so adjusted, that when 
light is admitted at -fiJ a pure spectrum \a formed at AB, the extremity of the 
long box. A mirror at M is also adjusted so as to reflect the light from E 
along the narrow compartment of the long box to BC. See Fig. 3. 

At ^5 is placed the contrivance shewn in Fig. 2, Plate I. ^'^ is a rect- 
angular frame of brass, having a rectangular aperture of 6 x 1 inches. On this 
frame are placed six brass sliders, A', Y, Z. Each of these carries a knife-edge 
of brass in the plane of the surface of the frame. 

These six moveable knife-edges form three sUts, X, Y, Z, which may be 
so adjusted as to coincide with any three portions of the pure spectrum formed 
by Hght from E. The intervals behind the sliders are closed by hinged shutters, 
which allow the sliders to move without letting hght pass between them. 

The inner edge of the brass frame is graduated to twentieths of an inch, 
so that the position of any slit can be read off. The breadth of the slit is 
ascertained by means of a wedge-shaped piece of metal, six inches long, and 
tapering to a point from a breadth of half an inch. This is gently inserted into 
each sht, and the breadth is determined by the distance to which it enters, the 
divisions on the wedge corresponding to the 200th of an inch difference in 
breadth, so that the unit of breadth is '005 inch. 

Now suppose hght to enter at E, to pass through the lens, and to be 
refracted by the two prisms at P; a pure spectrum, shewing Fraunhofer's lines, 
is formed at AB, but only that part is allowed to pass which faUs on the three 
slits X, Y, Z. The rest is stopped by the shutters. Suppose that the portion 
faUing on X belongs to the red part of the spectrum ; then, of the white Hght 
entering at E, only the red will come through the slit X. If we were to admit 
red Hght at X it would be refracted to E, by the principle in Optics, that the 
course of any ray may be reversed. If, instead of red light, we were to admit 
white light at X, still only red Hght would come to E ; for aU other light 
would be either more or less refracted, and would not reach the slit at E. 
Applying the eye at the slit E, we should see the prism P uniformly illuminated 
with red Hght, of the kind corresponding to the part of the spectrum which 
falls on the slit X when Hght is admitted at E. 

Let the sHt Y correspond to another portion of the spectrum, say the green ; 
then, if white light is admitted at Y, the prism, as seen by an eye at E, will 
be uniformly illuminated with green Hght; and if white Hght be admitted at X 



422 ON THE THEORY OF COMPOUND COLOURS. 

and Y simultaneously, tlie colour seen at E will be a compound of red and green, 
the proportions depending on the breadth of the sUts and the intensity of the 
Hght which enters them. The third sHt Z, enables us to combine any three kinds 
of light in any given proportions, so that an eye at E shall see the face of the 
prism at P uniformly illuminated with the colour resulting from the combination 
of the three. The position of these three rays in the spectrum is found by 
admitting the light at E, and comparing the position of the slits with the 
position of the principal fixed lines ; and the breadth of the sHts is determined 
by means of the wedge. 

At the same time white light is admitted through BC to the mirror of black 
glass at M, whence it is reflected to E, past the edge of the prism at P, so that 
the eye at E sees through the lens a field consisting of two portions, separated 
by the edge of the prism; that on the left hand being compounded of three 
colours of the spectrum refracted by the prism, while that on the right hand is 
white light reflected from the mirror. By adjusting the slits properly, these two 
portions of the field may be made equal, both in colour and brightness, so that 
the edge of the prism becomes almost invisible. 

In making experiments, the instrument was placed on a table in a room 
moderately lighted, with the end AB turned towards a large board covered with 
white paper, and placed in the open air, so as to be uniformly illuminated by 
the sun. In this way the thi'ee sHts and the mirror M were all illuminated 
with white light of the same intensity, and all were affected in the same ratio 
by any change of illumination; so that if the two halves of the field were 
rendered equal when the sun was under a cloud, they were found nearly correct 
when the sun again appeared. No experiments, however, were considered good 
unless the sun remained uniformly bright during the whole series of experiments. 

After each set of experiments light was admitted at E, and the position of 
the fixed lines D and F of the spectrum was read off on the scale at AB. It 
was found that after the instrument had been some time in use these positions 
were invariable, shewing that the eye-hole, the prisms, and the scale might be 
considered as rigidly connected. 



ON THE THEORY OF COMPOUND C0L0UBJ8. 423 



§ VI. Method of determining the Wave-length corresponding to any point 
of the Spectrum on the Scale AB. 

Two plane surfaces of glass were kept apart by two parallel strips of gold- 
beaters' leaf, so as to enclose a stratum of air of nearly uniform thickness. Light 
reflected from this stratum of air was admitted at E, and the spectrun formed 
by it was examined at AB by means of a lens. This spectrum consists of a 
large number of bright bands, separated by dark spaces at nearly uniform intervals, 
these intervals, however, being considerably larger as we approach the violet end 
of the spectrum. 

The reason of these alternations of brightness is easily explained. By the 
theory of Newton's rings, the light reflected from a stratum of air consists of 
two parts, one of which has traversed a path longer than that of the other, by 
an interval depending on the thickness of the stratum and the angle of incidence. 
Whenever the interval of retardation is an exact multiple of a wave-length, these 
two portions of light destroy each other by interference ; and when the interval 
is an odd number of half wave-lengths, the resultant light is a maximum. 

In the ordinary case of Newton's rings, these alternations depend upon the 
varying thickness of the stratum ; while in this case a pencil of rays of different 
wave-lengths, but aU experiencing the same retardation, is analysed into a spectrum, 
in which the rays are arranged in order of their respective wave-lengths. Every 
ray whose wave-length is an exact submultiple of the retardation will be destroyed 
by interference, and its place will appear dark in the spectrum; and there will 
be as many dark bands seen as there are rays whose wave-lengths ftdfil this 
condition. 

If, then, we observe the positions of the dark bands on the scale AB, 
tlie wave-lengths corresponding to these positions will be a series of submultiples 
of the retardation. 

Let us call the first dark band visible on the red side of the spectrum zero, 
and let us number them in order 1, 2, 3, &c. towards the violet end. Let N 
be the number of undulations corresponding to the band zero which are con- 
tained in the retardation R; then if n be the number of any other band, N+n 
wiU be the number of the corresponding wave-lengths in the retardation, or in 
symbols, 

R = (N+n)\ (6). 



424 ON THE THEORY OF COMPOUND COLOUBS. 

Now observe the position of two of Fraunhofer's fixed lines with respect to 
the dark bands, and let n„ n^ be their positions expressed in the number of 
bands, whole or fractional, reckoning from zero. Let Xj, X, be the wave-lengths 
of these fixed lines as determined by Fraunhofer, then 

R = (N+n,)K = (N+n,)K (7); 

whence N^-^^Jj^X^^n, (8), 

and -R = v' _ jj KK W 

Having thus found N and R, we may find the wave-length corresponding to 
the dark band n from the formula 

X = ^ (10). 

In my experiments the line D corresponded with the seventh dark band, and 
F was between the 15th and 16th, so that n^=15'7. Here then for D, 

. „ „ ^'~,'rr« ^~■.►rn.r '^ Fraunhofcr's measure (11), 

and for F, 7i,= 15-7, X,= 1794J "^ ' 

whence we find iV=34, i2 = 89175 (12). 

There were 22 bands visible, corresponding to 22 different positions on the 
scale AB, as determined 4th August, 1859. 



Table I. 



Band. 


Scale. 


Band. 


Scale. 


Band. 


Scale. 


n= 1 


17 


«= 9 


36 


n= 16 


57 


2 


19 


10 


39 


17 


61 


3 


21i 


11 


42 


18 


65 


4 


23J 


12 


45 


19 


69 


5 


26 


13 


48 


20 


73 


6 


28^ 


U 


51 


21 


77 


7 


31 


15 


54 


22 


82 


8 


33| 











Sixteen equidistant points on the scale were chosen for standard colours 
in the experiments to be described. The following Table gives the reading on 
the scale AB, the value of N+n, and the calculated wave-length for each of 
these : — 



ON THE THEORY OF COMPOUND COLOURS. 425 

Table II. 



oale. 


(N+«). 


Wave-length. 


Ck)lour. 


20 


36-4 


2450 


Red. 


24 


38-3 


2328 


Scarlet 


28 


39-8 


2240 


Orange. 


32 


41-4 


2154 


Yellow. 


36 


42-9 


2078 


Yellow-Green. 


40 


44-3 


2013 


Green. 


44 


45^7 


1951 


Green. 


48 


47-0 


1879 


Bluish green. 


52 


48-3 


1846 


Blue-green. 


56 


49-6 


1797 


Greenish blue. 


60 


50-8 


1755 


Blue. 


64 


51-8 


1721 


Blue. 


68 


52-8 


1688 


Blue. 


72 


53-7 


1660 


Indigo. 


76 


54-7 


1630 


Indigo. 


80 


55-6 


1604 


Indiga 



Having thus selected sixteen distinct points of the spectrum on which to 
operate, and determined their wave-lengths and apparent colours, I proceeded 
to ascertain the mathematical relations between these colours in order to lay 
them down on Newton's diagram. For this purpose I selected three of these 
as points of reference, namely, those at 24, 44, and 68 of the scale. I chose 
these points because they are weU separated from each other on the scale, and 
because the colour of the spectrum at these points does not appear to the eye 
to vary very rapidly, either in hue or brightness, in passing from one point to 
another. Hence a small error of position will not make so serious an alteration 
of colour at these points, as if we had taken them at places of rapid variation ; 
and we may regard the amount of the illumination produced by the light 
entering through the slits in these positions as sensibly proportional to the 
breadth of the slits. 

(24) corresponds to a bright scarlet about one-third of the distance from 
C to D; (44) is a green very near the line E; and (68) is a blue about one- 
third of the distance from F to G. 



42(3 ON THE THEORY OF COMPOUND COLOURS. 



§ VII. Method of Observation. 

The instrument is turned with the end AB towards a board, covered with 
white paper, and illuminated by sunlight. The operator sits at the end AB, to 
move the sliders, and adjust the sHts ; and the observer sits at the end E, 
which is shaded from any bright light. The operator then places the sHts so 
that their centres correspond to the three standard colours, and adjusts their 
breadths till the observer sees the prism iQuminated with pure white light of 
the same intensity with that reflected by the mirror M. In order to do this, 
the observer must tell the operator what difference he observes in the two halves 
of the illuminated field, and the operator must alter the breadth of the slits 
accordingly, always keeping the centre of each sKt at the proper point of the 
scale. The observer may call for more or less red, blue or green; and then 
the operator must increase or diminish the width of the slits X, Y, and Z 
respectively. If the variable field is darker or lighter than the constant field, 
the operator must Aviden or narrow all the slits in the same proportion. When 
the variable part of the field is nearly adjusted, it often happens that the 
constant white light from the mirror appears tinged with the complementary 
colour. This is an indication of what is required to make the resemblance of 
the two parts of the field of view perfect. When no difference can be detected 
between the two parts of the field, either in colour or in brightness, the observer 
must look away for some time, to relieve the strain on the eye, and then look 
again. If the eye thus refreshed still judges the two parts of the field to be 
equal, the observation may be considered complete, and the operator must measure 
the breadth of each slit by means of the wedge, as before described, and write 
down the result as a colour-equation, thus — 
Oct. 18, J. 18-5 (24) + 27 (44) + 37 (68) = W-^'^ (13). 

This equation means that on the 18th of October the observer J. (myself) made 
an observation in which the breadth of the slit X was 18-5, as measured by 
the wedge, while its centre was at the division (24) of the scale ; that the breadths 
of Y and Z were 27 and 37, and their positions (44) and (68) ; and that the 
illumination produced by these slits was exactly equal, in my estimation as an 
observer, to the constant white W. 



ON THE THEORY OF COMPOUND COLOURS. 427 

The position of 'the slit A" was then shifted from (24) to (28), and when 
the proper adjustments were made, I found a second colour-equation of this form — 
Oct. 18, J. 16 (28) + 21 (44) + 37 (68) = W (14). 

Subtracting one equation from the other and remembering that the figures in 
brackets are merely symbols of position, not of magnitude, we find 

16(28) = 18-5 (24) + 6(44) (15), 

shewing that (28) can be made up of (24) and (44), in the proportion of IS'o 
to 6. 

In this way, by combining each colour with two standard colours, we may 
produce a white equal to the constant white. The red and yellow colours from 
(20) to (32) must be combined with green and blue, the greens from (36) to (52) 
with red and blue, and the blues from (56) to (80) with red and green. 

The following is a specimen of an actual series of observations made in this 
way by another observer (K.) : — 

Table III. 

Oct. 13, 1859. Observer (K.). 

(X) {Y) {Z) 

18|(24) + 32^(44) + 32 (68) = W* 
17|(24) + 32|(44) + 63 (80) = W. 

18 (24) + 32|(44) + 35 (72) = W. 

19 (24) + 32 (44) + 31|(68) = W* 

19 (24) + 30|(44) + 35 (64) = W. 

20 (24) + 23 (44) + 39 (60) = W. 

21 (24) + 14 (44) + 58 (56) = W. 

22 (24) + 62 (52) + 11 (68) = W. 

22 (24) + 42 (48) + 29|(68) = W. 

19 (24) + 31|(44) + 33 (68) = W*. 

16 (24) + 28 (40) + 32^(68) = W. 
6 (24) + 27 (36) + 32^(68) = W. 

23 (32)+ 11|(44) + 821(68) = W. 

17 (28) + 26 (44) + 32^(68) = W. 

20 (24) + 33|(44) + 32|(68) = W». 
46 (20) + 33 (44) + 30 (68) = W. 

The equations marked with an asterisk (*') are those which involve the 
three standard colours, and since every other equation must be compared with 
them, they must be often repeated. 



428 ON THE THEORY OF COMPOUND COLOURS. 

The following Table contains the means of' four sets of observations by the 
same observer (K.) : — 

Table IV. (K.) 

44-3 (20) + 31 -0 (44) + 27-7 (68) = W. 
16-1 (28) + 25-6 (44) + 30-6 (68) = W. 
22-0 (32) + 12-1 (44) + 30-6 (68) = W. 
6-4 (24) + 25-2 (36) + 31 -3 (68) = W. 
15-3 (24) + 26 -0 (40) + 307 (68) = W. 
19-8 (24) + 35-0 (46) + 30-2 (68) = W. 
21-2 (24) + 41 -4 (48) ^ 27-0 (68) = W. 
22-0 (24) + 62-0 (52) + 13-0 (68) = W. 
21 -7 (24) + 10-4 (44) + 61 -7 (56) = W. 
20-5 (24) + 23-7 (44) + 40-5 (60) = W. 
19-7 (24) + 30-3 (44) + 33-7 (64) = W. 
18-0 (24) + 31-2 (44) + 32-3 (72) = W. 
17-5 (24) + 30-7 (44) + 44-0 (76) = W. 
18-3 (24) + 33-2 (44) + 63-7 (80) = W. 



§ VIII. Detet-mination of the Average Error in Observations of different kinds. 

In order to estimate the degree of accuracy of these observations, I have 
taken the differences between the values of the three standard colours as 
originally observed, and their means as given by the above Table. The sum 
of all the errors of the red (24) from the means, was 31 '1, and the number 
of observations was 42, which gives the average error 74. 

The sum of errors in green (44) was 48-0, and the number of observa- 
tions 31, giving a mean error 1-55. 

The sum of the errors in blue (68) was 46-9, and the number of observa- 
tions 35, giving a mean error 1*16. 

It appears therefore that in the observations generally, the average error 
does not exceed 1*5 ; and therefore the results, if confirmed by several obser- 
vations, may safely be trusted to that degree of accuracy. 

The equation between the three standard colours was repeatedly observed, 
in order to detect any alteration in the character of the light, or any other 
change of condition which would prevent the observations from being comparable 
with one another; and also because this equation is used in the reduction of 



(R)= -54 


(G-B)=-99 
(B-R)=-85 
(R-G)=-86 

G + B) = 2-67 


(G + B) = 2-31 
(B + R) = l-59 
(R + G) = l-57 


VG'4.B' = 


(G) = l-22 


JB' + R' 


(B) = M5 


jR' + ii'- 


(R + 


sfR' + G 


' + B^=l-76 



ON THE THEORY OF COMPOUND COLOURS. 429 

all the others, and therefore requires to be carefully observed. There are twenty 
observations of this equation, the mean of which gives 

18-6(24) + 31'4(44) + 30-5(68) = W* (16) 

as the standard equation. 

We may use the twenty observations of this equation as a means of 
determining the relations between the errors in the diflferent colours, and thus 
of estimating the accuracy of the observer in distinguishing colours. 

The following Table gives the result of these operations, where R stands 
for (24), G for (44), and B for (68):— 

Table V. — Mean Errors in the Standard Equation. 



1-26 
1-33 



The first column gives the mean difference between the observed value of 
each of the colours and the mean of all the observations. The second column 
shews the average error of the observed differences between the values of the 
standards, from the mean value of those differences. The third column shews 
the average error of the sums of two standards, from the mean of such sums. 
The fourth column gives the square root of the sum of the squares of the 
quantities in the first column. I have also given the average error of the 
sum of R, G and B, from its mean value, and the value of ^R^ + G' + B'. 

It appears from the first column that the red is more accurately observed 
than the green and blue. 

§ IX. Relative Accuracy in Observations of Colour and of Brightness. 

If the errors in the different colours occun^ed perfectly independent of each 
other, then the probable mean error in the sum or difference of any two colours 
would be the square root of the sum of their squares, as given in the fourth 
column. It will be seen, however, that the number in the second column is 
always less, and that in the third always greater, than that in the fourth ; 
shewing that the errors are not independent of each other, but that positive 
errors in any colour coincide more often with positive than with negative errors 



430 ON THE THEORY OF COMPOUND COLOURS. 

in another colour. Now the hue of the resultant depends on the ratios of the 
components, while its brightness depends on their sum. Since, therefore, the 
difference of two colours is always more accurately observed than their sum, 
variations of colour are more easily detected than variations in brightness, and 
the eye appears to be a more accurate judge of the identity of colour of the 
two parts of the field than of their equal illumiiiation. The same conclusion may 
be drawn from the value of the mean error of the sum of the three standards, 
which is 2-67, while the square root of the sum of the squares of the errors 
is 176. 

§ X. Reduction of the Observations. 

By eliminating W from the equations of page 428 by means of the standard 
equation, we obtain equations involving each of the fourteen selected colours of 
the spectrum, along with the three standard colours; and by transposing the 
selected colour to one side of the equation, we obtain its value in terms of 
the three standards. If any of the terms of these equations are negative, the 
equation has no physical interpretation as it stands, but by transposing the 
negative term to the other side it becomes positive, and then the equation may 
be verified. 

The following Table contains the values of the fourteen selected tints in 
terms of the standards. To avoid repetition, the symbols of the standard colours 
are placed at the head of each colunm. 





Table 


VI. 




Observer (K,). 


(24.) 


(44.) 


(68.) 


44-3(20) = 


18-6 


+ 0-4 


+ 2-8 


16-1(28) = 


18-6 


+ 5-8 


- 01 


22-0(32) = 


18-6 


+ 19-3 


- 01 


25-2(36) = 


12-2 


+ 31-4 


- 0-8 


26-0(40) = 


3-3 


+ 31-4 


- 0-2 


35-0(46) = 


- 1-2 


+ 31-4 


+ 0-3 


41-4(48) = 


- 2-6 


+ 31-4 


+ 3-5 


62-0(52) = 


- 3-4 


+ 31-4 


+ 17-5 


61-7(56) = 


- 3-1 


+ 21-0 


+ 30-5 


40-5(60) = 


- 1-9 


+ 7-7 


+ 30-5 


33-7(64) = 


- 11 


+ M 


+ 30-5 


32-3(72) = 


+ 0-6 


+ 0-2 


+ 30-5 


44-0(76) = 


+ 1-1 


+ 0-7 


+ 30-5 


63-7(80) = 


+ 0-3 


- 1-8 


+ 30-6 



ON THE THEORY OF COMPOUND COLOURS. 431 

From these equations we may lay down a chart of the spectrum on Newton's 
diagram by the following method : — Take any three points, A, B, C, and let A 
represent the standard colour (24), B (44), and C (68). Then, to find the position 
of any other colour, say (20), divide AC in P so that (18'6) ^P= (28) PC, and 
then divide BP in Q so that (IS'G + 2-8) P^ = (0-4) (?P. At the point Q the 
colour corresponding to (20) must be placed. In this way the diagram of fig. 4, 
Plate VI., p. 444, has been constructed from the observations of all the colours. 

§ XL Tlie Spectrum as laid down on Newton's Diagram. 

The curve on which these points lie has this striking feature, that two 
portions of it are nearly, if not quite, straight lines. One of these portions 
extends from (24) to (46), and the other from (48) to (64). The colour (20) 
and those beyond (64), are not far from the line joining (24) and (68). The 
spectrum, therefore, as exhibited in Newton's diagram, forms two sides of a 
triangle, with doubtful fi-agments of the third side. Now if three colours in 
Newton's diagram lie in a straight line, the middle one is a compound of the 
two others. Hence all the colours of the spectrum may be compounded of 
those which lie at the angles of this triangle. These correspond to the following 
colours : — 

Table VII. 







Scale. 


Wave-length. 


Index 

in water. 


Wave-length 
in water. 


R 


Scarlet . 


. 24 


2328 


1-332 


1-747 


G 


Green . . 


. 46f 


1914 


1-334 


1-435 


B 


Blue . . 


. 64i 


1717 


1-3.39 


1-282 



All the other colours of the spectrum may be produced by combinations of 
these; and since all natural colours are compounded of the colours of the spec- 
trum, they may be compounded of these three primary colours. I have strong 
reason to believe that these are the three primary colours corresponding to three 
modes of sensation in the organ of vision, on which the whole system of colour, 
as seen by the normal eye, depends. 

§ XII. Results found hy a second Observer. 

"We may now consider the results of three series of observations made by 
myself (J.) as observer, in order to determine the relation of one observer to 



432 



ON THE THEORY OF COMPOUND COLOURS. 



another in the perception of colour. The standard colours are connected by the 
following equation, as determined by six observations : — 

18-l(24) + 27-5(44) + 37(68) = W* (17). 

The average errors in these observations were — 



Table VIII. 



R, -28 
G, -83 
B, -16 



G + B, -83 
B + R, -42 

R + G, -95 



G - B, -83 
B-R, -28 
R-G, -72 



R + G + B, -95 



shewing that in this case, also, the power of distinguishing colour is more to be 
depended on than that of distinguishing degrees of illumination. 

The average error in the other observations from the means was '64 for red, 
76 for green, and 1*02 for blue. 



Table IX. 




Observations by 


J., October 


1859. 


(24.) 


(44.) 


(68.) 


44-3(20)= 18-1 


- 2-5 


+ 2-3 


16-0(28)= 18-1 


+ 6-2 


- 0-7 


21-5(32)= 18-1 


+ 25-2 


- 0-7 


19-3(36)= 8-1 


+ 27-5 


- 0-3 


20-7(40)= 2-1 


+ 27-5 


- 0-5 


52-3(48) = - 1-4 


+ 27-5 


^10-7 


95-0(52) = - 2-4 


+ 27-5 


+ 37-0 


51-7(56) = - 2-2 


+ 4-8 


+ 37-0 


37-2(60) = - 1-2 


+ 0-8 


+ 370 


36-7(64) = - 0-2 


+ 0-8 


+ 37-0 


350(72) = + 0-6 


- 0-2 


+ 37-0 


400(76) = + 0-9 


+ 0-5 


+ 37-0 


51-0(80) = + 1-1 


+ 0-5 


+ 37-0 



§ XIII. Comparison of Results hy Newton's Diagram. 

The relations of the colours, as given by these observations, are laid down 
in fig. 5, Plate VI., p. 444. It appears from this diagram, that the positions of 
the colours lie nearly in a straight line from (24) to (44), and from (48) to (60). 
The colours beyond (60) are crowded together, as in the other diagram, and 
the observations are not yet suflaciently accurate to distinguish their relative 
positions accurately. The coloiir (20) at the red end of the spectrum is further 



ON THE THEORY OF COMPOUND COLOURS. 433 

from tlie line joining (24) and (68) than in the other diagram, but I have not 
obtained satisfactory observations of these extreme colours. It will be observed 
that (32), (36), and (40) are placed further to the right in fig. 5 than in fig. 4, 
shewing that the second observer (J.) sees more green in these colours than 
the first (K.), also that (48), (52), (56), and (60) are much further up in fig. 5, 
shewing that to the second observer they appear more blue and less green. 
These differences were well seen in making an observation. When the instru- 
ment was adjusted to suit the first observer (K.), then, if the selected colour 
were (32), (36), or (40), the second (J.), on looking into the instrument, saw it 
too green ; but if (48), (52), (56), or (60) were the selected colour, then, if right 
to the first observer, it appeared too blue to the second. If the instrument 
were adjusted to suit the second observer, then, in the first case, the other saw 
red, and in the second green ; shewing that there was a real difference in the 
eyes of these two individuals, producing constant and measurable differences in 
the apparent colour of objects. 



§ XIV. Comparison hy Curves of Intensity of the Primaries. 

Figs. 6 and 7, Plate VI. p. 444, are intended to indicate the intensities of 
the three standard colours at different points of the spectrum. The curve marked 
(R) indicates the intensity of the red or (24), (G) that of green or (44), and (B) 
that of blue or (68). The curve marked (S) has its ordinates equal to the 
sum of the ordinates of the other three curves. The intensities are found by 
dividing every colour-equation by the coefficient of the colour on the left-hand 
side. Fig. 6 represents the results of observations by K., and fig. 7 represents 
those of J. It will be observed that the ordinates in fig. 7 are smaller between 
(48) and (56) than in fig. 6. This indicates the feeble intensity of certain kinds 
of light as seen by the eyes of J., which made it impossible to get observations 
of the colour (52) at all without making the slit so wide as to include all 
between (48) and (56). 

This blindness of my eyes to the parts of the spectrum between the fixed 
lines E and F appears to be confined to the region surrounding the axis of 
vision, as the field of view, when adjusted for my eyes looking directly at the 
colour, is decidedly out of adjustment when I view it by indirect vision, turning 
the axis of my eye towards some other point. The prism then appears greener 



434 ON THE THEORY OF COMPOUND COLOURS. 

and brighter than the mirror, shewing that the parts of my eye at a" distance 
from the axis are more sensitive to this blue-green light than the parts close 
to the axis. 

It is to be noticed that this insensibility is not to all light of a green 
or blue colour, but to Hght of a definite refrangibility. If I had a species of 
colour-blindness rendering me totally or partially insensible to that element of 
colour which most nearly corresponds with the light in question, then the light 
from the mirror, as well as that from the prism, would appear to me deficient 
in that colour, and I should still consider them chromatically identical ; or if 
there were any difierence, it would be the same for ail colours nearly the same 
in appearance, such as those just beyond the line F, which appear to me quite 
bright. 

We must also observe that the peculiarity is confined to a certain portion 
of the retina, which is known to be of a yellow colour, and which is the seat 
of several ocular phenomena observed by Purkinje and Wheatstone, and of the 
sheaf or brushes seen by Haidinger in polarized light ; and also that though, 
of the two observers whose results are given here, one is much more affected 
with this peculiarity than the other, both are less sensible to the light between 
E and F than to that on either side; and other observers, whose results are 
not here given, confirm this. 



§ XV. Explanation of the Differences between the two Observers. 

I think, therefore, that the yellow spot at the foramen centrale of Soemmering 
will be found to be the cause of this phenomenon, and that it absorbs the rays 
between E and F, and would, if placed in the path of the incident light, 
produce a corresponding dark band in the spectrum formed by a prism. 

The reason why white light does not appear yellow in consequence, is that 
this absorbing action is constant, and we reckon as white the mean of all the 
colours we are accustomed to see. This may be proved by wearing spectacles 
of any strong colour for some time, when we shall find that we judge white 
objects to be white, in spite of the rays which enter the eye being coloured. 

Now ordinary white light is a mixture of all kinds of light, including that 
between E and F, which is partially absorbed. If, therefore, we compound an 
artificial white containing the absorbed ray as one of its three components, it 



ON THE THEORY OF COMPOUND COLOURS. 435 

will be much more altered by the absorption than the ordinary light, which 
contains many rays of nearly the same colour, which are not absorbed. On the 
other hand, if the artificial light do not contain the absorbed ray, it will be 
less altered than the ordinary light which contains it. Hence the greater the 
absorption the less green will those colours appear which are near the absorbed 
part, such as (48), (52), (56), and the more green will the colours appear which 
are not near it, such a^ (32), (36), (40). And these are the chief differences 
between fig. 4 and fig. 5. 

I first observed this peculiarity of my eyes when observing the spectrum 
formed by a very long vertical slit. I saw an elongated dark spot running up 
and down in the blue, as if confined in a groove, and following the motion 
of the eye as it moved up or down the spectrum, but refusing to pass out 
of the blue into other colours. By increasing the breadth of the spectrum, the 
dark portion was found to correspond to the foramen centrale, and to be visible 
only when the eye is turned towards the blue-green between E and F. The 
spot may be well seen by first looking at a yellow paper, and then at a blue 
one, when the spot will be distinctly seen for a short time, but it soon dis- 
appears when the eye gets accustomed to the blue*. 

I have been the more careful in stating this peculiarity of my eyes, as I 
have reason to believe that it affects most persons, especially those who can see 
Haidinger's brushes easily. Such persons, in comparing their vision with that 
of others, may be led to think themselves affected with partial colour-blindness, 
whereas their colour-vision may be of the ordinary kind, but the rays which 
reach their sense of sight may be more or less altered in their proportions by 
passing through the media of the eye. . The existence of real, though partial 
colour-blindness will make itself apparent, in a series of observations, by the 
discrepancy between the observed values and the means being greater in certain 
colours than in others. 



§ XVI. General Conclusions. 

Neither of the observers whose results are given here shew any indications 
of colour-blindness, and when the differences arising from the absorption of the 
rays between E and F are put out of account, they agree in proving that there 
are three colours in the spectrum, red, green, and blue, by the mixtures of 

* See the Report of tlie British Association for 1856, p. 12. 



436 ON THE THEORY OF COMPOUND COLOURS. 

which colours chromatically identical with the other colours of the spectrum 
may be produced. The exact position of the red and blue is not yet ascer- 
tained; that of the green is ^ from E towards F. 

The orange and yellow of the spectrum are chromatically equivalent to 
mixtures of red and green. They are neither richer nor paler than the corre- 
sponding mixtures, and the only difference is that the mixture may be resolved 
by a prism, whereas the colour in the spectrum cannot be so resolved. This 
result seems to put an end to the pretension of yellow to be considered a 
primary element of colour. 

In the same way the colours from the primary green to blue are chro- 
matically identical with mixtures of these ; and the extreme ends of the spectrum 
are probably equivalent to mixtures of red and blue, but they are so feeble 
in illumination that experiments on the same plan with the rest can give no 
result, but they must be examined by some special method. When observations 
have been obtained from a greater number of individuals, including those whose 
vision is dichromatic, the chart of the spectrum may be laid down independently 
of accidental differences, and a more complete discussion of the laws of the 
sensation of colour attempted. 



POSTSCRIPT. 

[Keceived May 8,— Read May 24, I860.] 

Since sending the above paper to the Royal Society, I have obtained 
some observations of the colour of the spectrum by persons whose vision is 
"dichromic," and who are therefore said to be " colour-bhnd." 

The instrument used in making these observations was similar in principle 
to that formerly described, except that, in order to render it portable, the rays 
are reflected back through the prisms, nearly in their original direction ; thus 
rendering one of the limbs of the instrument unnecessary, and allowing the 
other to be shortened considerably on account of the greater angular dispersion. 
The principle of reflecting light, so as to pass twice through the same prism, 
was employed by me in an instrument for combining colours made in 1856, 
and a reflecting instrument for observing the spectrum has been constructed 
independently by M. Porro. 



ON THE THEORY OF COMPOUND COLOURS. 437 

Light from a sheet of paper illuminated by sunlight is admitted at the slits 
X, Y, Z (fig. 8, Plate VIL p. 444), falls on the prisms P and F (angles = 45"), 
then on a concave silvered glass, S, radius 34 inches. The light, after reflexion, 
passes again through the prisms R and P, and is reflected by a small mirror, 
e, to the slit E, where the eye is placed to receive the light compounded of 
the colours corresponding to the positions and breadths of the slits X, Y, and Z. 

At the same time, another portion of the light from the illuminated paper 
enters the instrument at BC, is reflected at the mirror M, passes through the 
lens L, is reflected at the mirror M', passes close to the edge of the prism P, 
and is reflected along with the coloured light at e, to the eye-slit at E. 

In this way the compound colour is compared with a constant white light 
in optical juxtaposition with it. The mirror M is made of silvered glass, that 
at M' is made of glass roughened and blackened at the back, to reduce the 
intensity of the constant light to a convenient value for the experiments. 

This instrument gives a spectrum in which the lines are very distinct, 
and the length of the spectrum from A to H is, 3-6 mches. The outside 
measure of the box is 3 feet 6 inches, by 11 inches by 4 inches, and it can 
be carried about, and set up in any position, without readjustment. It was 
made by Messrs Smith and Ramage of Aberdeen. 

In obtaining observations from colour-blind persons, two sHts only are 
required to produce a mixture chromatically equivalent to white; and at one 
point of the spectrum the colour of the pure rays appears identical with white. 
This point is near the line F, a little on the less refrangible side. From this 
point to the more refrangible end of the spectrum appears to them "blue." 
The colours on the less refrangible side appear to them all of the same quahty, 
but of different degrees of brightness; and when any of them are made 
sufficiently bright, they are called "yellow." It is convenient to use the term 
"yellow" in speaking of the colours from red to green inclusive, since it will 
be found that a dichromic person in speaking of red, green, orange, and brown, 
refers to different degrees of brightness or purity of a single colour, and not 
to different colours perceived by him. This colour we may agree to call 
"yellow," though it is not probable that the sensation of it is like that of 
yellow as perceived by us. 

Of the three standard colours which I formerly assumed, the red appears 
to them "yellow," but so feeble that there is not enough in the whole red 
division of the spectrum to form an equivalent to make up the standard white. 



438 ON THE THEORY OF COMPOUND COLOURS. 

The green at E appears a good "yellow," and the blue at f from F towards 
G appears a good "blue." I have therefore taken these as standard colours for 
reducing dichromic observations. The three standard colours will be referred to 
as (104), (88), and (68), these being the positions of the red, green, and blue on 
the scale of the new instrument. 

Mr James Simpson, formerly student of Natural Philosophy in my class, has 
ftimished me with thirty-three observations taken in good sunlight. Ten of 
these were between the two standard colours, and give the following result : — 

337 (88) + 33-1 (68) = W (1). 

The mean errors of these observations were as follows : — 

Error of (88) = 2-5; of (68) = 2-3; of (88) + (68) = 4'8 ; of (88)-(68) = 1-3. 

The fact that the mean error of the sum was so much greater than the mean 
error of the difference indicates that in this case, as in all others that I have 
examined, observations of equality of tint can be depended on much more than 
observations of equality of illumination or brightness. 

From six observations of my own, made at the same time, I have deduced 
the " trichromic " equation 

22-6 (104)4-26 (88) + 37-4 (68) = W (2). 

If we suppose that the light which reached the organ of vision was the 
same in both cases, we may combine these equations by subtraction, and so find 
22-6(104)-77(88) + 4-3(68) = i> (3), 

where D is that colour, the absence of the sensation of which constitutes the 
defect of the dichromic eye. The sensation which I have in addition to 
those of the dichromic eye is therefore similar to the full red (104), but 
different from it, in that the red (104) has 7'7 of green (88) in it which must 
be removed, and 4*3 of blue (68) substituted. This agrees pretty well with the 
colour which Mr Pole* describes as neutral to him, though crimson to others. 
It must be remembered, however, that different persons of ordinary vision require 
different proportions of the standard colours, probably owing to differences in the 
absorptive powers of the media of the eye, and that the above equation (2), if 
observed by K., would have been 

23(104) + 32(88) + 3l(68) = W (4). 

♦ Philosophical Transactions, 1859, Part I. p. 329. 



ON THE THEORY OF COMPOUND COLOURS. 



439 



and the value of D, as deduced from these observers, would have been 

23(104)- 17 (88)- ri (68) = Z) (5), 

in which the defective sensation is much nearer to the red of the spectrum. It 
is probably a colour to which the extreme red of the spectrum tends, and 
which differs from the extreme red only in not containing that small proportion 
of "yellow" light which renders it visible to the colour-blind. 

From other observations by Mr Simpson the following results have been 
deduced : — 



Table a. 





(88.) 


(68.) 


(99-2 + ) = 


337 


1-9 


31-3(96) = 


33-7 


2-1 


28 (92) = 


33-7 


1-4 


33-7(88) = 


33-7 





54-7(84) = 


33-7 


6-1 


71 (82) = 


33-7 


15-1 


99 (80) = 


33-7 


33-1 


70 (78) = 


15-7 


33-1 


56 (76) = 


5-7 


331 


36 (72) = 


- 0-3 


33-1 


33-1(68) = 





33-1 


40 (64) = 


0-2 


33-1 


55-5(60) = 


1-7 


33-1 


(57-) = 


- 0-3 


33-1 





(88.) 


(68.) 


100(96) = 


108 


7 


100(92) = 


120 


6 


100(88) = 


100 





100(84) = 


61 


11 


100(82) = 


47 


21 


100(80) = 


34 


33 


100(78) = 


22 


47 


100(76) = 


10 


59 


100(72) = 


- 1 


92 


100(68) = 





100 


100(64) = 





83 


100(60) = 


3 


60 



In the Table on the left side (99*2 + ) means the whole of the spectrum beyond 
(99'2) on the scale, and (57-) means the whole beyond (57) on the scale. The 
position of the fixed lines with reference to the scale was as follows : — 

A, 116; a, 112; B, 110; C, 106; D, 98-3; E, 88; F, 79; G, 61; H, 44. 

The values of the standard colours in different parts of the spectrum are given 
on the right side of the above Table, and are represented by the cai-ves of 
fig. 9, Plate VII. p. 444, where the left-hand curve represents the intensity 
of the "yellow" element, and the right-hand curve that of the "blue" element 
of colour as it appears to the colour-blind. 

The appearance of the spectrum to the colour-blind is as follows: — 
From A to E the colour is pure " yellow " very faint up to D, and 
reaching a maximum between D and E. From E to one-third beyond F towards 



440 ON THE THEORY OF COMPOUND COLOURS. 

G the colour is mixed, varying from " yellow " to " blue," and becoming neutral 
or "white" at a point near F. In this part of the spectrum, the total inten- 
sity, as given by the dotted line, is decidedly less than on either side of it, and 
near the line F, the retina close to the "yellow spot" is less sensible to light 
than the parts further from the axis of the eye. This peculiarity of the light 
near F is even more marked in the colour-blind than in the ordinary eye. 
Beyond F the " blue " element comes to a maximum between F and G, and 
then diminishes towards H ; the spectrum from this maximum to the end being 
pure "blue." 

In fig. 10, Plate VII. p. 444, these results are represented in a different 
manner. The point D, corresponding to the sensation wanting in the colour-blind, 
is taken as the origin of coordinates, the "yellow" element of colour is represented 
by distances measured horizontally to the right from D, and the "blue" element 
by distances measured vertically from the horizontal line through D. The 
numerals indicate the different colours of the spectrum according to the scale 
shewn in fig. 9, and the coordinates of each point indicate the composition of 
the corresponding colour. The triangle of colours is reduced, in the case of 
dichromic vision, to a straight line "B" "Y," and the proportions of "blue" 
and "yellow" in each colour are indicated by the ratios in which this line is 
cut by the line from D passing through the position of that colour. 

The results given above were all obtained with the light of white paper, 
placed in clear simshine. I have obtained similar results, when the sun was 
hidden, by using the light of uniformly illuminated clouds, but I do not consider 
these observations suflficiently free from disturbing circumstances to be employed 
in calculation. It is easy, however, by means of such observations, to verify the 
most remarkable phenomena of colour-blindness, as for instance, that the colours 
from red to green appear to differ only in brightness, and that the brightness 
may be made identical by changing the width of the slit; that the colour 
near F is a neutral tint, and that the eye in viewing it sees a dark spot in 
the direction of the axis of vision ; that the colours beyond are all blue of 
different intensities, and that any "blue" may be combined with any "yellow" 
in such proportions as to form "white." These results I have verified by the 
observations of another colour-blind gentleman, who did not obtain sunlight for 
bis observations; and as I have now the means of carrying the requisite 
apparatus easily, I hope to meet with other colour-blind observers, and to obtain 
their observations under more favourable circumstances. 



ON THE THEORY OF COMPOUND COLOURS. 441 



On the Comparison of Colour-blind with ordinary Vision by means of Observations 

with Coloured Papers. 

In March 1859 I obtained a set of observations by Mr Simpson, of the 
relations between six coloured papers as seen by him. The experiments were 
made with the colour-top in the manner described in my paper in the Trans- 
actions of the Royal Society of Edinburgh, Vol. xxi. pt. 2, p. 286; and the 
colour-equations were arranged so as to be equated to zero, as in those given 
in the Philosophical Magazine, July, 1857. The colours were — Vermilion (V), 
ultramarine (U), emerald-green (G), ivory-black (B), snow-white (W), and pale 
chrome-yellow (Y). These six colours afford fifteen colour-blind equations, since 
four colours enter into each equation. Fourteen of these were observed by 
Mr Simpson, and from these I deduced three equations, giving the relation of 
the three standards (V), (U), (G) to the other colours, according to his kind of 
vision. From these three equations I then deduced fifteen equations, admitting 
of comparison with the observed equations, and necessarily consistent in 
themselves. 

The comparison of these equations furnishes a test of the truth of the theory 
that the colour-blind see by means of two colour-sensations, and that therefore 
eveiy colour may be expressed in terms of two given colours, just as in ordinary 
vision it may be expressed in terms of three given colours. The one set of 
equations are each the result of a single observation ; the other set are deduced 
from three equations in accordance with this theory, and the two sets agree to 
within an average error = 2*1. 

Table b. 







V. 


U. 


G. 


B. 


W. 


Y. 


1. 


Observed . . 








-100 


+ 45 


+ 22 


+ 33 =0. 




Calculated . 








-100 


+ 37-5 


+ 26-5 


+ 36 =0. 


2. 


Observed . . 





+ 58 





-69 


-31 


-42 =0. 




Calculated . 





+ 58-3 





-67-3 


-32-7 


+ 41-7 = 0. 


3. 


Observed . . 





+ 32 


-100 





+ 12 


+ 56 =0. 




Calculated . 





+ 32-3 


-100 





+ 8-3 


+ 59-4 = 0. 


4. 


Observed . . 





+ 38 


- 89 


-11 





+ 62 =0. 




Calculated . 





+ 40 


- 85 


-15 





+ 60 =0. 


5. 


Observed . . 





+ 32 


+ 68 


-60 


-40 


-0. 




Calculated . 





+ 34 


+ 66 


-63-5 


-36-5 


=0. 



442 ON THE THEORY OF COMPOUND COLOURS. 

Table b (continued). 







V. 


U. 




G. 


B. 


W. 


Y. 


6. 


Observed . . 


.-100 










+ 82 


+ 5 


+ 13 =0. 




Calculated . 


.-100 










+ 83-9 


+ 4-5 


+ 11-6 = 0. 


7. 


Observed . . 


.+ 47 





- 


100 





+ 22 


+ 31 = 0. 




Calculated . 


.+ 44-7 





- 


100 





+ 24-5 


+ 30-8 = 0. 


8. 


Observed . . 


.-100 





+ 


20 


+ 77 





+ 3 =0. 




Calculated . 


.-100 





+ 


17 


+ 77-5 





+ 5-5=0. 


9. 


Not Observed 
















Calculated . 


.+ 96 





- 


31 


-69 


+ 4 


=0. 


10. 


Observed . . 


.- 70 


+ 53 










-30 


+ 47 =0. 




Calculated . 


.- 73-5 


+ 53 










-26-5 


+ 47 =0. 


11. 


Observed . . 


.-100 


+ 8 







+ 71 





+ 21 =0. 




Calculated . 


.-100 


+ 8 







+ 74-5 





+ 17-5 = 0. 


12. 


Observed . . 


.+ 85 


+ 15 







-88 


-12 


=0. 




Calculated . 


.+ 86 


+ 14 







-88-5 


-11-5 


=0. 


13. 


Observed . . 


.- 20 


+ 39 


_ 


80 








+ 61 =0. 




Calculated . 


.- 19 


+ 40 


- 


81 








+ 60 =0. 


14. 


Observed . . 


.- 66 


+ 30 


+ 


70 





-34 


=0. 




Calculated . 


.- 70 


+ 27 


+ 


73 





-30 


=0. 


15. 


Observed . . 


. + 100 


- 2 


_ 


27 


-71 





=0. 




Calculated . 


.+ 96 


+ 4 


- 


24 


-76 





=0. 



But, axjcording to our theory, colour-blind vision is not only dichromic, but 
the two elements of colour are identical with two of the three elements of 
colour as seen by the ordinary eye ; so that it differs from ordinary vision 
only in not perceiving a particular colour, the relation of which to known colours 
may be numerically defined. This colour may be expressed under the form 

aV + 6U + cG = D (16), 

where V, U, and G are the standard colours used in the experiments, and D is 
the colour which is visible to the ordinary eye, but invisible to the colour- 
blind. If we know the value of D, we may always change an ordinary colour- 
equation into a colour-blind equation by subtracting from it nD (n being chosen 
so that one of the standard colours is eliminated), and adding n of black. 

In September 1856 I deduced, from thirty-six observations of my own, the 
chromatic relations of the same set of six coloured papers. These observations, 
with a comparison of them with the trichromic theory of vision, are to be 
found in the Philosophical Magazine for July 1857. The relations of the 



ON THE THEORY OF COMPOUND COLOURS. 443 

six colours may be deduced from two equations, of which the most convenient 

form is 

V. U. G. B. W. Y. 

+ 397 +2G-6 +337 -227 -77-3 =0 (17). 

-62-4 +18-6 -37-6 +457 +357 = (18). 

The value of D, as deduced from a comparison of these equations with the 
colour-blind equations, is 

1-198 V + 0-078U-0-276G = D (19). 

By making D the same thing as black (B), and eliminating W and Y 
respectively from the two ordinary colour-equations by means of D, we obtain 
three colour-blind equations, calculated from the ordinary equations and con- 
sistent with them, supposing that the colour (D) is black to the colour-blind. 

The following Table is a comparison of the colour-bhnd equations deduced 
from Mr Simpson's observations alone, with those deduced from my observations 
and the value of D. 







Table 


C. 










V. 


u. 


G. 


B. 


w. 


Y. 


(15) Calculated 


. +96 


+ 4 


-24 


-76 








By (19) . . . 


. +93-9 


+ 6-1 


-21-7 


-78-3 








(U) Calculated 


. -70 


+ 27 


+ 73 





-30 





By (17) and (19) 


. -70 


+ 27-2 


-72-8 





-30 





(13) Calculated 


. -19 


+ 40 


-81 








+ 60 


By (18) and (19) 


. -13-6 


+ 38-5 


-86-4 








+ 61-5 



The average error here is 1*9, smaller than the average error of the indi- 
vidual colour-blind observations, shewing that the theory of colour-blindness being 
the want of a certain colour-sensation which is one of the three ordinary colour- 
sensations, agrees with observation to within the limits of error. 

In fig. 11, Plate VII. p. 444, I have laid down the chromatic relations of these 
colours according to Newton's method. V (vermilion), U (ultramarine), and G 
(emerald-green) are assumed as standard colours, and placed at the angles of 
an equilateral triangle. The position of W (white) and Y (pale chrome-yellow) 
with respect to these are laid down from equations (17) and (18), deduced 
from my own observations. The positions of the defective colour, of white, and 
of yellow, as deduced from Mr Simpson's equations alone, are given at " c7," 
"w" and "y." The positions of these points, as deduced from a combination 



444 ON THE THEORY OF COMPOUND COLOURS. 

» 

of these equations with my o\7n, are given at "D," *'W," and "Y." The 
difference of these positions from those of "c?," "w;," and "3/," shews the amount 
of discrepancy between observation and theory. 

It will be observed that D is situated near V (vermilion), but that a line 
from D to W cuts UV at C near to V. D is therefore a red colour, not 
scarlet, but further from yellow. It may be called crimson, and may be imitated 
by a mixture of 86 vermiHon and 14 ultramarine. This compound colour will be 
of the same hue as D ; but since C hes between D and W, C must be 
regarded as D diluted with a certain amount of white ; and therefore D must 
be imagined to be like C in hue, but without the intermixture of white which 
is unavoidable in actual pigments, and which reduces the purity of the tint. 

Lines drawn from D through "W" and "Y," the colour-blind positions of 
white and yeUow, pass through W and Y, their positions in ordinary vision. 
The reason why they do not coincide with W and Y, is that the white and 
yeUow papers are much brighter than the colours corresponding to the points 
W and Y of the triangle V, U, G; and therefore lines from D, which represent 
them in intensity as well as in quality, must be longer than DW and DY in 
the proportion of their brightness. 



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VOL. I. PLATE VII. (II) 





[Lecture at the Royal Institution of Great Britain. May 17, 1861.] 



XXII. On the Theory of Three Primary Colours. 

The speaker commenced by shewing that our power of vision depends 
entirely on our being able to distinguish the intensity and quality of colours. 
The forms of visible objects are indicated to us only by differences in colour 
or brightness between them and surrounding objects. To classify and arrange 
these colours, to ascertain the physical conditions on which the dijfferences of 
coloured rays depend, and to trace, as far as we are able, the physiological 
process by which these different rays excite in us various sensations of colour, 
we must avail ourselves of the united experience of paintei-s, opticians, and 
physiologists. The speaker then proceeded to state the results obtained by these 
three classes of inquirers, to explain their apparent inconsistency by means of 
Young's Theory of Primary Colours, and to describe the tests to which he had 
subjected that theory. 

Painters have studied the relations of colours, in order to imitate them by 
means of pigments. As there are only a limited number of coloured substances 
adapted for painting, while the number of tints in nature is infinite, painters 
are obliged to produce the tints they require by mixing their pigments in 
proper proportions. This leads them to regard these tints as actually com- 
pounded of other colours, corresponding to the pure pigments in the mixture. 
It is found, that by using three pigments only, we can produce all colours 
lying within certain limits of intensity and purity. For instance, if we take 
carmine (red), chrome yellow, and ultramarine (blue), we get by mixing the 
carmine and the chrome, all varieties of orange, passing through scarlet to 
crimson on the one side, and to yeUow on the other; by mixing chrome and 
ultramarine we get all hues of green; and by mixing ultramarine with carmine, 
we get all hues of purple, from violet to mauve and crimson. Now these are 
all the strong colours that we ever see or can imagine : all others are like 



446 ON THE THEORY OF THREE PRIMARY COLOURS. 

these, only less pure in tint. Our three colours can be mixed so as to form 
a neutral grey; and if this grey be mixed with any of the hues produced by 
mixing two colours only, all the tints of that hue will be exhibited, from the 
pure colour to neutral grey. If we could assume that the colour of a mixture 
of different kinds of paint is a true mixture of the colours of the pigments, 
and in the same proportion, then an analysis of colour might be made with 
the same ease as a chemical analysis of a mixture of substances. 

The colour of a mixture of pigments, however, is often very different from 
a true mixture of the colours of the pure pigments. It is found to depend on 
the size of the particles, a finely ground pigment producing more effect than 
one coarsely ground. It has also been shewn by Professor Helmholtz, that when 
light falls on a mixture of pigments, part of it is acted on by one pigment 
only, and part of it by another ; while a third portion is acted on by both pig- 
ments in succession before it is sent back to the eye. The two parts reflected 
directly from the pure pigments enter the eye together, and form a true mixture 
of colours ; but the third portion, which has suffered absorption from both 
pigments, is often so considerable as to give its own character to the resulting 
tint. This is the explanation of the green tint produced by mixing most blue 
and yellow pigments. 

In studying the mixture of colours, we must avoid these sources of error, 
either by mixing the rays of light themselves, or by combining the impressions 
of colours within the eye by the rotation of coloured papers on a disc. 

The speaker then stated what the opticians had discovered about colour. 
White light, according to Newton, consists of a great number of different kinds 
of coloured light which can be separated by a prism. Newton divided these 
into seven classes, but we now recognize many thousand distinct kinds of light 
in the spectrum, none of which can be shewn to be a compound of more 
elementary rays. If we accept the theory that light is an undulation, then, 
as there are undulations of every different period from the one end of the 
spectrum to the other, there are an infinite number of possible kinds of Hght, 
no one of which can be regarded as compounded of any others. 

Physical optics does not lead us to any theory of three primary colours, 
but leaves us in possession of an infinite number of pure rays with an infinitely 
more infinite number of compound beams of Hght, each containing any propor- 
tions of any number of the pure rays. 

These beams of light, passing through the transparent parts of the eye, fall 



ON THE THEORY OF THREE PRIMARY COLOURS. 447 

on a sensitive membrane, and we become aware of various colours. We know 
that the colour we see depends on the nature of the light; but the opticians 
say there are an infinite number of kinds of light ; while the painters, and all 
who pay attention to what they see, tell us that they can account for all 
actual colours by supposing them mixtures of three primary colours. 

The speaker then next drew attention to the physiological difficulties in 
accounting for the perception of colour. Some have supposed that the different 
kinds of light are distinguished by the time of their vibration. There are 
about 447 billions of vibrations of red light in a second; and 577 billions of 
vibrations of green light in the same time. It is certainly not by any mental 
process of which we are conscious that we distinguish between these infini- 
tesimal portions of time, and it is difficult to conceive any mechanism by which 
the vibrations could be counted so that we should become conscious of the 
results, especially when many rays of different periods of vibration act on the 
same part of the eye at once. 

Besides, all the evidence we have on the nature of nervous action goes 
to prove that whatever be the nature of the agent which excites a nerve, the 
sensation will differ only in being more or less acute. By acting on a nerve 
in various ways, we may produce the faintest sensation or the most violent 
pain ; but if the intensity of the sensation is the same, its quality must be 
the same. 

Now, we may perceive by our eyes a faint red light which may be made 
stronger and stronger till our eyes are dazzled. We may then perform the 
same experiment with a green light or a blue light. We shall thus see that 
our sensation of colour may differ in other ways, besides in being stronger or 
fainter. The sensation of colour, therefore, cannot be due to one nerve only. 

The speaker then proceeded to state the theory of Dr Thomas Young, as 
the only theory which completely reconciles these difficulties in accounting for 
the perception of colour. 

Young supposes that the eye is provided with three distinct sets of nervous 
fibres, each set extending over the whole sensitive surface of the eye. Each 
of these three systems of nerves, when excited, gives us a different sensation. 
One of them, which gives us the sensation we call red, is excited most by 
the red rays, but also by the orange and yellow, and slightly by the violet ; 
another is acted on by the green rays, but also by the orange and yellow and 
part of the blue; while the third is acted on by the blue and violet rays. 



448 ON THE THEORY OF THREE PRIMARY COLOURS. 

If we could excite one of these sets of nerves without acting on the 
others, we should have the pure sensation corresponding to that set of nerves. 
This would be truly a primary colour, whether the nerve were excited by pure 
or by compound light, or even by the action of pressure or disease. 

If such experiments could be made, we should be able to see the primary- 
colours separately, and to describe their appearance by reference to the scale 
of colours in the spectrum. 

But we have no direct consciousness of the contrivances of our own bodies, 
and we never feel any sensation which is not infinitely complex, so that we 
can never know directly how many sensations are combined when we see a 
colour. Still less can we isolate one or more sensations by artificial means, so 
that in general when a ray enters the eye, though it should be one of the 
pure rays of the spectrum, it may excite more than one of the three sets of 
nerves, and thus produce a compound sensation. 

The terms simple and compound, therefore, as applied to colour-sensation, 
have by no means the same meaning as they have when appHed to a ray of 
light. 

The speaker then stated some of the consequences of Young's theory, and 
described the tests to which he had subjected it: — 

1st. There are three primary colours. 

2nd. Every colour is either a primary colour, or a mixture of primary 
colours. 

3rd. Four colours may always be arranged in one of two ways. Either 
one of them is a mixture of the other three, or a mixture of two of them 
can be found, identical with a mixture of the other two. 

4th. These results may be stated in the form of colour-equations, giving 
the numerical value of the amount of each colour entering into any mixture. 
By means of the Colour Top'", such equations can be obtained for coloured 
papers, and they may be obtained with a degree of accuracy shewing that the 
colour-judgment of the eye may be rendered very perfect. 

The speaker had tested in this way more than 100 different pigments and 
mixtures, and had found the results agree with the theory of three primaries 

* Described in the Trans, of the Royal Society of Edinburgh, Vol. xxi., and in the Phil. Mag. 



ON THE THEORY OF THREE PRIMARY COLOURS. 449 

in every case. He had also examined all the colours of the spectrum with 
the same result. 

The experiments with pigments do not indicate what colours are to be 
considered as primary ; but experiments on the prismatic spectrum shew that 
all the colours of the spectrum, and therefore all the colours in nature, are 
equivalent to mixtures of three colours of the spectrum itself, namely, red, 
green (near the line E), and blue (near the line G). Yellow was found to be 
a mixture of red and green. 

The speaker, assuming red, green, and blue as primary colours, then exhi- 
bited them on a screen by means of three magic lanterns, before which were 
placed glass troughs containing respectively sulphocyanide of iron, chloride of 
copper, and ammoniated copper. 

A triangle was thus illuminated, so that the pure colours appeared at its 
angles, while the rest of the triangle contained the various mixtures of the 
colours as in Young's triangle of colour. 

The graduated intensity of the primary colours in different parts of the 
spectrum was exhibited by three coloured images, which, when superposed on 
the screen, gave an artificial representation of the spectrum. 

Three photographs of a coloured ribbon taken through the three coloured 
solutions respectively, were introduced into the camera, giving images represent- 
ing the red, the green, and the blue parts separately, as they would be seen 
by each of Young's three sets of nerves separately. When these were super- 
posed, a coloured image was seen, which, if the red and green images had 
been as fully photographed as the blue, would have been a truly-coloured image 
of the ribbon. By finding photographic materials more sensitive to the less 
refrangible rays, the representation of the colours of objects might be greatly 
improved. 

The speaker then proceeded to exhibit mixtures of the colours of the pure 
spectrum. Light from the electric lamp was passed through a narrow slit, a 
lens and a prism, so as to throw a pure spectrum on a screen containing three 
moveable slits, through which three distinct portions of the spectrum were 
suffered to pass. These portions were concentrated by a lens on a screen at 
a distance, forming a large, uniformly coloured image of the prism. 

When the whole spectrum was allowed to pass, this image was white, as 
in Newton's experiment of combining the rays of the spectrum. When portions 
of the spectrum were allowed to paas through the moveable slits, the image was 

VOL. L 57 



450 ON THE THEORY OF THREE PRIMARY COLOURS. 

uniformly illuminated with a mixture of the corresponding colours. In order 
to see these colours separately, another lens was placed between the moveable 
slits and the screen. A magnified image of the sHts was thus thrown on the 
screen, each sHt shewing, by its colour and its breadth, the quality and quantity 
of the colour which it suffered to pass. Several colours were thus exhibited, 
first separately, and then in combination. Red and blue, for instance, produced 
purple ; red and green produced yellow ; blue and yellow produced a pale pink ; 
red, blue, and green produced white; and red and a bluish green near the 
line F produced a colour which appears very different to different eyes. 

The speaker concluded by stating the peculiarities of colour-blind vision, 
and by shewing that the investigation into the theory of colour is truly a 
physiological inquiry, and that it requires the observations and testimony of 
persons of every kind in order to discover and explain the various peculiarities 
of vision. 



[From the Philosophical Magazine, Vol. xxi.] 



XXIII. On Physical Lines of Force. 

PART I. 
The Theory of Molecular Vortices applied to Magnetic Phenomena. 

In all phenomena involving attractions or repulsions, or any forces depend- 
ing on the relative position of bodies, we have to determine the magnitude and 
direction of the force which would act on a given body, if placed in a given 
position. 

In the case of a body acted on by the gravitation of a sphere, this force 
is inversely as the square of the distance, and in a straight line to the centre 
of the sphere. In the case of two attracting spheres, or of a body not spherical, 
the magnitude and direction of the force vary according to more complicated 
laws. In electric and magnetic phenomena, the magnitude and direction of the 
resultant force at any point is the main subject of investigation. Suppose that 
the direction of the force at any point is known, then, if we draw a line so 
that in every part of its course it coincides in direction with the force at that 
point, this hne may be called a line of force, since it indicates the direction 
of the force in every part of its course. 

By drawing a sufficient number of lines of force, we may indicate the 
direction of the force in every part of the space in which it acts. 

Thus if we strew iron filings on paper near a magnet, each filing will be 
magnetized by induction, and the consecutive filings will unite by their opposite 
poles, so as to form fibres, and these fibres will indicate the direction of the lines 
of force. The beautiful illustration of the presence of magnetic force afforded 
by this experiment, naturally tends to make us think of the lines of force as 
something real, and as indicating something more than the mere resultant of 
two forces, whose seat of action is at a distance, and which do not exist there 

57—2 



452 ON PHYSICAL LINES OF FORCE. 

at all until a magnet is placed in that part of the field. We are dissatisfied 
with the explanation founded on the hypothesis of attractive and repellent 
forces directed towards the magnetic poles, even though we may have satisfied 
ourselves that the phenomenon is in strict accordance with that hypothesis, and 
we cannot help thinking that in every place where we find these lines of force, 
some physical state or action must exist in sufficient energy to produce the 
actual phenomena. 

My object in this paper is to clear the way for speculation in this direction, 
by investigating the mechanical results of certain states of tension and motion 
in a medium, and comparing these with the observed phenomena of magnetism 
and electricity. By pointing out the mechanical consequences of such hypotheses, 
I hope to be of some use to those who consider the phenomena as due to the 
action of a medium, but are in doubt as to the relation of this hypothesis to 
the experimental laws already established, which have generally been expressed 
in the language of other hypotheses. 

I have in a former paper* endeavoured to lay before the mind of the 
geometer a clear conception of the relation of the lines of force to the space 
in which they are traced. By making use of the conception of currents in a 
fluid, I shewed how to draw lines of force, which should indicate by their 
number the amount of force, so that each line may be called a unit-line of 
force (see Faraday's Reswear dies, 3122); and I have investigated the path of 
the lines where they pass from one medium to another. 

In the same paper I have found the geometrical significance of the "Elec- 
trotonic State," and have shewn how to deduce the mathematical relations 
between the electrotonic state, magnetism, electric currents, and the electromotive 
force, using mechanical illustrations to assist the imagination, but not to account 
for the phenomena. 

I propose now to examine magnetic phenomena from a mechanical point of 
view, and to determine what tensions in, or motions of, a medium are capable 
of producing the mechanical phenomena observed. If, by the same hypothesis, 
we can connect the phenomena of magnetic attraction with electromagnetic phe- 
nomena and with those of induced currents, we shall have found a theory 
which, if not true, can only be proved to be erroneous by experiments which 
will greatly enlarge our knowledge of this part of physics. 

♦ See a paper " On Faraday's Lines of Force," Cambridge Philosophical Transactions, Vol. i. Part i. 
Page 155 of this volume. 



ON PHYSICAL LINES OF FORCE. 453 

The mechanical conditions of a medium under magnetic influence have been 
variously conceived of, as currents, undulations, or states of displacement or 
strain, or of pressure or stress. 

Currents, issuing from the north pole and entering the south pole of a 
magnet, or circulating round an electric current, have the advantage of repre- 
senting correctly the geometrical arrangement of the lines of force, if we could 
account on mechanical principles for the phenomena of attraction, or for the 
currents themselves, or explain their continued existence 

Undulations issuing from a centre would, according to the calculations of 
Professor Challis, produce an effect similar to attraction in the direction of the 
centre ; but admitting this to be true, we know that two series of undulations 
traversing the same space do not combine into one resultant as two attractions 
do, but produce an effect depending on relations of phase as well as intensity, 
and if allowed to proceed, they diverge from each other without any mutual 
action. In fact the mathematical laws of attractions are not analogous in any 
respect to those of undulations, while they have remarkable analogies with those 
of currents, of the conduction of heat and electricity, and of elastic bodies. 

In the Cambridge and Dublin Mathematical Journal for January 1847, 
Professor William Thomson has given a "Mechanical Representation of Electric, 
Magnetic, and Galvanic Forces," by means of the displacements of the particles of 
an elastic solid in a state of strain. In this representation we must make the 
angular displacement at every point of the solid proportional to the magnetic 
force at the con-esponding point of the magnetic field, the direction of the axis 
of rotation of the displacement corresponding to the direction of the magnetic 
force. The absolute displacement of any particle will then correspond in magni- 
tude and direction to that which I have identified with the electrotonic state ; 
and the relative displacement of any particle, considered with reference to the 
particle in its immediate neighbourhood, will correspond in magnitude and direc- 
tion to the quantity of electric current passing through the corresponding point 
of the magneto-electric field. The author of this method of representation does 
not attempt to explain the origin of the observed forces by the effects due to 
these strains in the elastic solid, but makes use of the mathematical analogies 
of the two problems to assist the imagination in the study of both. 

We come now to consider the magnetic influence as existing in the form of 
some kind of pressure or tension, or, more generally, of stress in the medium. 

Stress is action and reaction between the consecutive parts of a body, and 



454 ON PHYSICAL LINES OF FORCE. 

consists in general of pressures or tensions different in different directions at 
the same point of the medium. 

The necessary relations among these forces have been investigated by mathe- 
maticians ; and it has been shewn that the most general type of a stress 
consists of a eombmation of three principal pressures or tensions, in directions 
at right angles to each other. 

When two of the principal pressures are equal, the third becomes an axis 
of symmetry, either of greatest or least pressure, the pressures at right angles 
to this axis being all equal. 

When the three principal pressures are equal, the pressure is equal in every 
direction, and there results a stress having no determinate axis of direction, of 
which we have an example in simple hydrostatic pressure. 

The general type of a stress is not suitable as a representation of a mag^ 
netic force, because a line of magnetic force has direction and intensity, but 
has no third quahty indicating any difference between the sides of the line, 
which would be analogous to that observed in the case of polarized light*. 

We must therefore represent the magnetic force at a point by a stress 
having a single axis of greatest or least pressure, and all the pressures at right 
angles to this axis equal. It may be objected that it is inconsistent to represent 
a line of force, which is essentially dipolar, by an axis of stress, which is 
necessarily isotropic; but we know that every phenomenon of action and reaction 
is isotropic in its results, because the effects of the force on the bodies between 
which it acts are equal and opposite, while the nature and origin of the force 
may be dipolar, as in the attraction between a north and a south pole. 

Let us next consider the mechanical effect of a state of stress symmetrical 
about an axis. We may resolve it, in all cases, into a simple hydrostatic 
pressure, combined with a simple pressure or tension along the axis. When the 
axis is that of greatest pressure, the force along the axis will be a pressure. 
When the axis is that of least pressure, the force along the axis will be a 
tension. 

K we observe the lines of force between two magnets, as indicated by iron 
filings, we shall see that whenever the Hnes of force pass firom one pole to 
another, there is attraction between those poles; and where the lines of force 
from the poles avoid each other and are dispersed into space, the poles repel 

* See Faraday's Researches, 3262. 



ON PHYSICAL LINES OF FORCE. 455 

each other, so that in both cases they are drawn in the direction of the 
resultant of the lines of force. 

It appears therefore that the stress in the axis of a line of magnetic force 
is a tension, like that of a rope. 

If we calculate the lines of force in the neighbourhood of two gravitating 
bodies, we shall find them the same in direction as those near two magnetic 
poles of the same name ; but we know that the mechanical effect is that of 
attraction instead of repulsion. The lines of force in this case do not run 
between the bodies, but avoid each other, and are dispersed over space. In 
order to produce the effect of attraction, the stress along the lines of gravi- 
tating force must be a pressure. 

Let us now suppose that the phenomena of magnetism depend on the 
existence of a tension in the direction of the lines of force, combined with a 
hydrostatic pressure; or in other words, a pressure greater in the equatorial 
than in the axial direction : the next question is, what mechanical explanation 
can we give of this inequality of pressures in a fluid or mobUe medium ? The 
explanation which most readily occurs to the mind is that the excess of pres- 
sure in the equatorial direction arises from the centrifugal force of vortices or 
eddies in the medium having their axes in directions parallel to the lines of force. 

This explanation of the cause of the inequality of pressures at once suggests 
the means of representing the dipolar character of the line of force. Every 
vortex is essentially dipolar, the two extremities of its axis being distinguished 
by the direction of its revolution as observed from those points. 

We also know that when electricity circulates in a conductor, it produces 
lines of magnetic force passing through the circuit, the direction of the lines 
depending on the direction of the circulation. Let us suppose that the direction 
of revolution of our vortices is that in which vitreous electricity must revolve 
in order to produce lines of force whose direction within the circuit is the 
same as that of the given lines of force. 

We shall suppose at present that all the vortices in any one part of the 
field are revolving in the same direction about axes nearly parallel, but 
that in passing from one part of the field to another, the direction of the 
axes, the velocity of rotation, and the density of the substance of the vortices 
are subject to change. We shall investigate the resultant mechanical effect upon 
an element of the medium, and from the mathematical expression of this 
resultant we shall deduce the physical character of its different component parts. 



456 ON PHYSICAL LINES OF FORCE. 

Prop. I. — If in two fluid systems geometrically similar the velocities and 
densities at corresponding points are proportional, then the differences of pres- 
sure at corresponding points due to the motion will vary in the duplicate ratio 
of the velocities and the simple ratio of the densities. 

Let I be the ratio of the linear dimensions, m that of the velocities, 
n that of the densities, and p that of the pressures due to the motion. Then 
the ratio of the inasses of corresponding portions will be Vn, and the ratio of 
the velocities acquired in traversing similar parts of the systems will be m ; 
so that l^mn is the ratio of the momenta acquired by similar portions in 
traversing similar parts of their paths. 

The ratio of the surfaces is P, that of the forces acting on them is I'^p, 

and that of the times during which they act is — ; so that the ratio of the 
impulse of the forces is — , and we have now 

m 
or m^n =jp ; 

that is, the ratio of the pressures due to the motion (p) is compounded of 
the ratio of the densities (n) and the duplicate ratio of the velocities {ni"), and 
does not depend on the linear dimensions of the moving systems. 

In a circular vortex, revolving with uniform angular velocity, if the 
pressure at the axis is p^, that at the circumference will be i>i=jPo + ip^j where 
p is the density and v the velocity at the circumference. The mean pressure 
parallel to the axis will be 

If a number of such vortices were placed together side by side with their 
axes parallel, they would form a medium in which there would be a pressure 
Pz parallel to the axes, and a pressure p^ in any perpendicular direction. If the 
vortices are circular, and have uniform angular velocity and density throughout, 
then 

Pi-P2 = lp'^' 
If the vortices are not circular, and if the angular velocity and the density 
are not uniform, but vary according to the same law for all the vortices, 

Pi-p.^Cpif, 



ON PHYSICAL LINES OF FORCE. 457 

where p is the mean density, and C is a numerical quantity depending on the 
distribution of angular velocity and density in the vortex. In future we shall 

write -7^ instead of Co, so that 

477- '^ 

^'"^'^4^''''' (^)' 

where /n is a quantity bearing a constant ratio to the density, and v is the 
linear velocity at the circumference of each vortex. 

A medium of this kind, filled with molecular vortices having their axes 
parallel, differs from an ordinary fluid in having different pressures in different 
directions. If not prevented by properly arranged pressures, it would tend to 
expand laterally. In so doing, it would allow the diameter of each vortex to 
expand and its velocity to diminish in the same proportion. In order that a 
medium having these inequalities of pressure in different directions should be in 
equihbrium, certain conditions must be fulfilled, which we must investigate. 

Prop. II. — If the direction-cosines of the axes of the vortices with respect 
to the axes of x, y, and z be /, m, and n, to find the normal and tangential 
stresses on the co-ordinate planes. 

The actual stress may be resolved into a simple hydrostatic pressure p^ acting 
in all directions, and a simple tension Pi—p^, or -7- fiif, acting along the axis 
of stress. 

Hence if p^x, pyy, and p^ be the normal stresses parallel to the three axes, 
considered positive when they tend to increase those axes ; and if p^^, p^, and 
Pj^ be the tangential stresses in the three co-ordinate planes, considered positive 
when they tend to increase simultaneously the symbols subscribed, then by 
the resolution of stresses*, 

Pxx = j^l^vn'-p„ 
1 . , 

* Rankine's Applied Mechanics, Art. 106. 
VOL. I. 58 



458 



ON PHYSICAL LINES OF FORCE. 



If we write 
then 



a = vl, ^ = vm, and y = vn, 






Air 
1 






(2). 



Prop. III. — To find the resultant force on an element of the medium, 
arising from the variation of internal stress. 

"We have in general, for the force in the direction of x per unit of volume 
by the law of equilibrium of stresses*, 

V d d d ,„v 

^'TxP-'+TyP-' + dzP' (^)- 

In this case the expression may be written 

Remembering that a ^ + /8 ^ + y ^ = i ^ (a" + jff + y"), this becomes 

. I ld& da.\ _ 1 Ida. dy\ dp, , , 

-l'^i^[di-Ty)+l'->'Tn[di-di)-dS---^^'- 

The expressions for the forces parallel to the axes of y and z may be written 
down from analogy. 

* Baiikine's Applied MecJianics, Art. 116. 



ON PHYSICAL LINES OF FORCE. 459 

We have now to interpret the meaning of each term of this expression. 
We suppose a, /3, y to be the components of the force which would act 
upon that end of a unit magnetic bar which points to the north. 

/x represents the magnetic inductive capacity of the medium at any point 
referred to air as a standard, /la, /i,/3, /xy represent the quantity of magnetic 
induction through unit of area perpendicular to the three axes of x, y z 
respectively. 

The total amount of magnetic induction through a closed surface surrounding 
the pole of a magnet, depends entirely on the strength of that pole ; so that 
if dxdydz be an element, then 

(-T-/xa + -i-/>t/3 + -T- /lyj dxdydz = i'rrm dxdydz (6), 

which represents the total amount of magnetic induction outwards through the 
surface of the element dxdydz, represents the amount of "imaginary magnetic 
matter" within the element, of the kind which points north. 

The first term of the value of X, therefore, 

1 /d d n d \ /_. 

''ii[dx''^ + d^l'^ + dz''V (^)' 

may be written 

am (8), 

where a is the intensity of the magnetic force, and m is the amount of mag- 
netic matter poLnting north in unit of volume. 

The physical interpretation of this term is, that the force urging a north pole 
in the positive direction of a; is the product of the intensity of the magnetic 
force resolved in that direction, and the strength of the north pole of the magnet. 

Let the parallel lines from left to right in fig. 1 represent a field of mag- 
netic force such as that of the earth, sn being the direction from south to north. 
The vortices, according to our hypothesis, will be in the direction shewn by the 
arrows in fig. 3, that is, in a plane perpendicular to the lines of force, and 
revolving in the direction of the hands of a watch when observed from 5 
looking towards n. The parts of the vortices above the plane of the paper 
will be moving towards e, and the parts below that plane towards w. 

58—2 



460 



ON PHYSICAL LINES OF FORCE. 



Fig. 1. 



1^ 


^ 


\ k/ 




(^c r 


"^ ^TJ 


"ll 


t<: ^ 


1 ^^ 




/ X 


N 


' " 




€ 

Fig. 2. 

1 




V 


/ 


^ 


\n 


'^ 




S B >@-^ Y 


.^ 


/• 


^\ 




/ 


\ 


^ 



We shall always mark by an arrow-head the direction in which we must 
look in order to see the vortices rotating in the 
direction of the hands of a watch. The arrow-head 
will then indicate the northward direction in the 
magnetic field, that is, the direction in which that 
end of a magnet which points to the north would 
set itself in the field. 

Now let A be the end of a magnet which 
points north. Since it repels the north ends of 
other magnets, the Hues of force wiU be directed 
from A outwards in all directions. On the north 
side the line AD wiU be in the sarae direction with 
the lines of the magnetic field, and the velocity of 
the vortices will be increased. On the south side 
the line AC will be in the opposite direction, and 
the velocity of the vortices wUl be diminished, so 
that the lines of force are more powerful on the 
north side of A than on the south side. 

We have seen that the mechanical efiect of the 
vortices is to produce a tension along their axes, 
so that the resultant effect on A will be to pull 

it more powerfully towards D than towards C\ that is, A will tend to move 
to the north. 

Let B in fig. 2 represent a south pole. The lines of force belonging to B 
will tend towards B, and we shall find that the lines of force are rendered 
stronger towards E than towards F, so that the effect in this case is to urge B 
towards the south. 

It appears therefore that, on the hypothesis of molecular vortices, our first 
term gives a mechanical explanation of the force acting on a north or south 
pole in the magnetic field. 

We now proceed to examine the second term, 

Here a^' + ^ + y* is the square of the intensity at any part of the field, and 
ft, is the magnetic inductive capacity at the same place. Any body therefore 




ON PHYSICAL LINES OF FORCE, 4GI 

placed in the field will be urged towards places of stronger magnetic intensity 
with a force depending partly on its own capacity for magnetic induction, and 
partly on the rate at which the square of the intensity increases. 

If the body be placed in a fluid medium, then the medium, as well as the 
body, will be urged towards places of greater intensity, so that its hydrostatic 
pressure will be increased in that direction. The resultant effect on a body 
placed in the medium will be the difference of the actions on the body and 
on the portion of the medium which it displaces, so that the body will tend 
to or from places of greatest magnetic intensity, according as it has a greater 
or less capacity for magnetic induction than the surrounding medium. 

In fig. 4 the lines of force are represented as converging and becoming 
more powerful towards the right, so that the magnetic tension at B is stronger 
than at A, and the body AB will be urged to the right. If the capacity for 
magnetic induction is greater in the body than in the surrounding medium, it 
will move to the right, but if less it will move to the left. 

Fig. 4. Fig. 5. 




We may suppose in this case that the lines of force are converging to a 
magnetic pole, either north or south, on the right hand. 

In fig. 5 the Hues of force are represented as vertical, and becoming more 
numerous towards the right. It may be shewn that if the force increases 
towards the right, the lines of force will be curved towards the right. The 
effect of the magnetic tensions wiU then be to draw any body towards the right 
with a force depending on the excess of its inductive capacity over that of the 
surrounding medium. 

We may suppose that in this figure the lines of force are those surrounding 
an electric current perpendicular to the plane of the paper and on the right 
hand of the figure. 

These two iUustrations will shew the mechanical effect on a paramagnetic 
or diamagnetic body placed in a field of varying magnetic force, whether the 
increase of force takes place along the lines or transverse to them. The form 



462 ON PHYSICAL LINES OF FORCE. 

of the second term of our equation indicates the general law, which is quite 
independent of the direction of the lines of force, and depends solely on the 
manner in which the force varies from one part of the field to another. 

"We come now to the third term of the value of X, 

1 fd/B da.\ 



^^ 47r \dx dy, 

Here y^^ is, as before, the quantity of magnetic induction through unit of area 
perpendicular to the axis of y, and -J- — -j- ^^ ^ quantity which would disap- 
pear if adx + ^dy + ydz were a complete differential, that is, if the force acting 
on a unit north pole were subject to the condition that no work can be done 
upon the pole in passing round any closed curve. The quantity represents the 
work done on a north pole in travelHng round unit of area in the direction 
from +x to +y parallel to the plane of xy. Now if an electric current whose 
strength is r is traversing the axis of z, which, we may suppose, points 
vertically upwards, then, if the axis of x is east and that of y north, a unit 
north pole will be urged round the axis of z in the direction from x to y, so 

that in one revolution the work done will be = 47rr. Hence t- ( -t^ — 7- ) repre- 



477 \dy 



Att \dx dy/ 

sents the strength of an electric current parallel to z through unit of area ; and 
if we write 

dz] P' 4,w\dz dx)~^- 4n\dx dyj~^ ^^'' 

then p, q, r will be the quantity of electric current per unit of area perpen- 
dicular to the axes of x, y, and z respectively. 

The physical interpretation of the third term of X, —fi^r, is that if /xyS is 
the quantity of magnetic induction parallel to y, and r the quantity of electricity 
flowing in the direction of z, the element will be urged in the direction of —x, 
transversely to the direction of the current and of the lines of force; that is, 
an ascending current in a field of force magnetized towards the north would 
tend to move west. 

To illustrate the action of the molecular vortices, let sn be the direction 
of magnetic force in the field, and let C be the section of an ascending mag- 
netic current perpendicular to the paper. The lines of force due to this current 



ON PHYSICAL LINES OF FORCE. 



463 



will be circles drawn in the opposite direction from that of the hands of a 

watch ; that is, in the direction nwse. At c the lines of force 

will be the sum of those of the field and of the current, and 

at w they will be the difference of the two sets of lines ; so 

that the vortices on the east side of the current will be more 

powerful than those on the west side. Both sets of vortices have 

their equatorial parts turned towards C, so that they tend to 

expand towards C, but those on the east side have the greatest 

effect, so that the resultant effect on the current is to urge it towards the west 

The fourth term, 

^da dy 



Fig. 6. 



1 da 



or ^-iiyq 



(10), 



may be interpreted in the same way, and indicates that a current q in the 
direction of y, that is, to the north, placed in a magnetic field in which the 
lines are vertically upwards in the direction of z, will be urged towards the ecLnt. 

The fifth term, 

dx 



(n), 



merely implies that the element wiQ be urged in the direction in which the 
hydrostatic pressure p^ diminishes. 

We may now write down the expressions for the components of the resultant 
force on an element of the medium per unit of volume, thus : 



^"^"^^^ ^('^)"''^'' + ''>'^"^ (^^)' 






fiyp + n-tar — 



dp, 
dy 



(13), 



The first term of each expression refers to the force acting on magnetic 



poles. 



The second term to the action on bodies capable of magnetism by induction. 
The third and fourth terms to the force acting on electric currents. 
And the fifth to the effect of simple pressure. 



464 ON PHYSICAL LINES OF FORCE. 

Before going further in the general investigation, we shall consider equations 
(12, 13, 14), in particular cases, corresponding to those simplified cases of the 
actual phenomena which we seek to obtain in order to determine their laws by 
experiment. 

We have found that the quantities p, q, and r represent the resolved parts 
of an electric current in the three co-ordinate directions. Let us suppose in the 
first instance that there is no electric current, or that p, q, and r vanish. We 
have then by (9), 

^_^ = ^-^ = ^-^ = (15) 

dy dz ' dz dx ' dx dy ^ '' 

whence we learn that adx + /3dy + ydz = d<l) (16), 

is an exact differential of <^, so that 

-t ^ = f • r = f (m: 

fi is proportional to the density of the vortices, and represents the " capacity 
for magnetic induction" in the medium. It is equal to 1 in air, or in whatever 
medium the experiments were made which determined the powers of the magnets, 
the strengths of the electric currents, &c. 

Let us suppose fi constant, then 



m 



=h{T>'^^4^^^^4M--rA?^^9^'^) (-) 



represents the amount of imaginary magnetic matter in unit of volume. That 
there may be no resultant force on that unit of volume arising from the action 
represented by the first term of equations (12, 13, 14), we must have m = 0, or 

'J-g^-S = o (-)• 

Now it may be shewn that equation (19), if true within a given space, 
implies that the forces acting within that space are such as would result from 
a distribution of centres of force beyond that space, attracting or repelling 
inversely as the square of the distance. 

Hence the lines of force in a part of space where fi is uniform, and where 
there are no electric currents, must be such as would result from the theory 
of "imaginary matter" acting at a distance. The assumptions of that theory 
are unlike those of ours, but the results are identical 



dr LL r' ^ '' 



ON PHYSICAL LINES OF FORCE. 465 

Let us first take the case of a single magnetic pole, that is, one end of 
a long magnet, so long that its other end is too far off to have a perceptible 
influence on the part of the field we are considering. The conditions then are, 
that equation (18) must be fulfilled at the magnetic pole, and (19) everywhere 
else. The only solution under these conditions is 

't'=--,l (^«). 

where r is the distance from the pole, and m the strength of the pole. 
The repulsion at any point on a unit pole of the same kind is 

d(f> _'in 1 

In the standard medium /i = 1 ; so that the repulsion is simply — in that 

medium, as has been shewn by Coulomb. 

In a medium having a greater value of fi (such as oxygen, solutions of 
salts of iron, &c.) the attraction, on our theory, ought to be less than in air, 
and in diamagnetic media (such as water, melted bismuth, &c.) the attraction 
between the same magnetic poles ought to be greater than in air. 

The experiments necessary to demonstrate the difference of attraction of two 
magnets according to the magnetic or diamagnetic character of the medium in 
which they are placed, would require great precision, on account of the limited 
range of magnetic capacity in the fluid media known to us, and the small 
amount of the difference sought for as compared with the whole attraction. 

Let us next take the case of an electric current whose quantity is C, 
flowing through a cylindrical conductor whose radius is R, and whose length is 
infinite as compared with the size of the field of force considered. 

Let the axis of the cylinder be that of z, and the direction of the current 
positive, then within the conductor the quantity of current per unit of area is 

C 1 /d^ da\ 



) (22): 



ir-R* Air \dx dy^ 
80 that within the conductor 

o-=-2^,y, /3 = 2-^a:, y = (23). 

VOL. L 59 



466 ON PHYSICAL LINES OF FORCE. 

Beyond the conductor, in the space round it, 

«^ = 2Ctan-' ^ (24), 

« = i=-^^^.- ^ = g = ^^^-^.' r = f = (25). 

If p — sjdi^-^y^ is the perpendicular distance of any point from the axis of 
the conductor, a unit north pole will experience a force = — , tending to move 

it round the conductor in the direction of the hands of a watch, if the observer 
view it in the direction of the current. 

Let us now consider a current running parallel to the axis of z in the 
plane of xz at a distance p. Let the quantity of the current be c', and let 

the length of the part considered be I, and its section 5, so that - is its 

strength per unit of section. Putting this quantity for p in equations (12, 13, 
14), we find 

^= -M^ "- 

per unit of volume; and multiplying by Is, the volume of the conductor con- 
sidered, we find 

X= -p.^c'1 

= -2.f (26), 

shewing that the second conductor will be attracted towards the first with a 
force inversely as the distance. 

We find in this case also that the amount of attraction depends on the 

value of /A, but that it varies directly instead of inversely as /i ; so that the 

attraction between two conducting wires will be greater in oxygen than in air, 
and greater in air than in water. 

We shall next consider the nature of electric currents and electromotive 
forces in connexion with the theory of molecular vortices. 



ON PHYSICAL LINES OF FORCE. 467 

PART 11. 
The Theory of Molecular Vortices applied to Electric Currents. 

We have already shewn that all the forces acting between magnets, sub- 
stances capable of magnetic induction, and electric currents, may be mechanically 
accounted for on the supposition that the surrounding medium is put into such 
a state that at every point the pressures are different in different directions, 
the direction of least pressure being that of the observed lines of force, and 
the difference of greatest and least pressures being proportional to the square 
of the intensity of the force at that point. 

Such a state of stress, if assumed to exist in the medium, and to be 
arranged according to the known laws regulating lines of force, will act upon 
the magnets, currents, &c. in the field with precisely the same resultant forces 
as those calculated on the ordinary hypothesis of direct action at a distance. 
This is true independently of any particular theory as to the cause of this 
state of stress, or the mode in which it can be sustained in the medium. We 
have therefore a satisfactory answer to the question, "Is there any mechanical 
hypothesis as to the condition of the medium indicated by lines of force, by 
which the observed resultant forces may be accounted for?" The answer is, 
the hues of force indicate the direction of minimum pressure at every point of 
the medium. 

The second question must be, "What is the mechanical cause of this 
difference of pressure in different directions?" We have supposed, in the first 
part of this paper, that this difference of pressures is caused by molecular 
vortices, having their axes parallel to the lines of force. 

We also assumed, perfectly arbitrarily, that the direction of these vortices 
is such that, on looking along a line of force from south to north, we should 
see the vortices revolving in the direction of the hands of a watch. 

We found that the velocity of the circumference of each vortex must be 
proportional to the intensity of the magnetic force, and that the density of 
the substance of the vortex must be proportional to the capacity of the medium 
for magnetic induction. 

We have as yet given no answers to the questions, " How are these vortices 
set in rotation?" and "Why are they arranged according to the known laws 

59—2 



468 ON PHYSICAL LINES OF FORCE. 

of lines of force about magnets and currents?" These questions are certainly 
of a higher order of difficulty than either of the former ; and I wish to separate 
the suggestions I may offer by way of provisional answer to them, from the 
mechanical deductions which resolved the first question, and the hypothesis of 
vortices which gave a probable answer to the second. 

We have, in fact, now come to inquire into the physical connexion of these 
vortices with electric currents, while we are still in doubt as to the nature of 
electricity, whether it is one substance, two substances, or not a substance at 
all, or in what way it differs from matter, and how it is connected with it. 

We know that the lines of force are affected by electric currents, and we 
know the distribution of those lines about a current ; so that from the force 
we can determine the amount of the current. Assuming that our explanation 
of the lines of force by molecular vortices is correct, why does a particular 
distribution of vortices indicate an electric current? A satisfactory answer to 
this question would lead us a long way towards that of a very important one, 
"What is an electric current?" 

I have found great difficulty in conceiving of the existence of vortices in a 
medium, side by side, revolving in the same direction about parallel axes. The 
contiguous portions of consecutive vortices must be moving in opposite directions ; 
and it is difficult to understand how the motion of one part of the medium 
can coexist with, and even produce, an opposite motion of a part in contact 
with it. 

The only ibnception which has at all aided me in conceiving of this kind of 
motion is that of the vortices being separated by a layer of particles, revolving 
each on its own axis in the opposite direction to that of the vortices, so that 
the contiguous surfaces of the particles and of the vortices have the same 
motion. 

In mechanism, when two wheels are intended to revolve in the same direc- 
tion, a wheel is placed between them so as to be in gear with both, and this 
wheel is called an "idle wheel." The hypothesis about the vortices which I 
have to suggest is that a layer of particles, acting as idle wheels, is interposed 
between each vortex and the next, so that each vortex has a tendency to make 
the neighbouring vortices revolve in the same direction with itself 

In mechanism, the idle wheel is generally made to rotate about a fixed 
axle; but in epicyclic trains and other contrivances, as, for instance, in Siemens's 



ON PHYSICAL LINES OF FORCE. 469 

governor for steam-engines*, we find idle wheels whose centres are capable of 
motion. In all these cases the motion of the centre is the half sum of the 
motions of the circumferences of the wheels between which it is placed. Let 
us examine the relations which must subsist between the motions of our vortices 
and those of the layer of particles interposed as idle wheels between them. 

Prop. IV. — To determine the motion of a layer of particles separating two 
vortices. 

Let the circumferential velocity of a vortex, multiplied by the three direc- 
tion-cosines of its axis respectively, be a, ;8, y, as in Prop. II. Let I, m, n be 
the direction- cosines of the normal to any part of the surface of this vortex, 
the outside of the surface being regarded positive. Then the components of the 
velocity of the particles of the vortex at this part of its surface will be 

nfi — my parallel to x, 

hf — na parallel to y, 

ma — l^ parallel to z. 

If this portion of the surface be in contact with another vortex whose velocities 
are a, ^, y, then a layer of very small particles placed between them will 
have a velocity which wiU be the mean of the superficial velocities of the 
vortices which they separate, so that if u ia the velocity of the particles in 
the direction of x, 

u = ^m(y-y)^in{^-fi) (27), 

since the normal to the second vortex is in the opposite direction to that of 
the first. 

Prop. V. — To determine the whole amount of particles transferred across 
unit of area in the direction of x in unit of time. 

Let Xi, 2/1, Zi be the co-ordinates of the centre of the first vortex, x.,, y„, z., 
those of the second, and so on. Let F,, Fj, &c. be the volumes of the first, 
second, &c. vortices, and F the sum of their volumes. Let dS be an element 
of the surface separating the first and second vortices, and x, y, z its co-ordinates. 
Let p be the quantity of particles on every unit of surface. Then if p be the 
whole quantity of particles transferred across irnit of area in unit of time in 

♦ See Goodeve's ElemenU of Mechanism, p. 118. 



470 ON PHYSICAL LINES OF FORCE. 

the direction of rr, the whole momentum parallel to x of the particles within 
the space whose volume is V will be Fp, and we shall have 

Vp==tupdS (28), 

the summation being extended to every surface separating any two vortices 
within the volume V. 

Let us consider the surface separating the first and second vortices. Let an 
element of this surface be dS, and let its direction-cosines be Zj, m^, n^^ with 
respect to the first vortex, and l^, m^, n, with respect to the second; then we 
know that 

^1 + 4 = 0, mi + ma = 0, ni + n, = (29). 

The values of a, ^, y vary with the position of the centre of the vortex ; 
so that we may write 

with similar equations for )8 and y. 

The value of u may be written >— 

w = i ^ H {x-x,) + m^ (x-x,)] 

+i^H(2/-2/i)+w2(2/-y.)}+i^H (2-^0+^.(2-2;.)} 

-l-J^{^i{^-^^) + '^h{x-x,)]-:^-£j{n,{y-y,) + n,{y-y,)] 

-if K(2-2.) + n, (.-.,)} (31). 

In effecting the summation of %updS, we must remember that round any 
closed surface XldS and all similar terms vanish ; also that terms of the form 
XlydS, where I and y are measured in different directions, also vanish; but that 
terms of the form tlxdS, where I and x refer to the same axis of co-ordinates, 
do not vanish, but are equal to the volume enclosed by the surface. The 
result is 



^^=4''(|-S<'''+''"+*") ^''^' 



ON PHYSICAL LINES OF FORCE. 471 

or dividing by F= F,+ F,4-&c., 



i^l-f) '^^)- 



If we make P = 7r (3^). 

then equation (33) will be identical with the first of equations (9), which give 
the relation between the quantity of an electric current and the intensity of 
the lines of force surrounding it. 

It appears therefore that, according to our hypothesis, an electric current 
is represented by the transference of the moveable particles interposed between 
the neighbouring vortices. We may conceive that these particles are very small 
compared with the size of a vortex, and that the mass of all the particles 
together is inappreciable compared with that of the vortices, and that a great 
many vortices, with their surrounding particles, are contained in a single complete 
molecule of the medium. The particles must be conceived to roll without sliding 
between the vortices which they separate, and not to touch each other, so that, 
as long as they remain within the same complete molecule, there is no loss of 
energy by resistance. When, however, there is a general transference of par- 
ticles in one direction, they must pass from one molecule to another, and in 
doing so, may experience resistance, so as to waste electrical energy and generate 
heat. 

Now let us suppose the vortices arranged in a medium in any arbitraiy 

manner. The quantities j^ — ~r > &c. will then in general have values, so that 

there will at first be electrical currents in the medium. These will be opposed 
by the electrical resistance of the medium ; so that, unless they are kept up 
by a continuous supply of force, they will quickly disappear, and we shall then 

have j^ "~ ;j~ = ^> ^^•'> ^^^^ is, adx + fidy + ydz will be a complete difierential 

(see equations (15) and (16)); so that our hypothesis accounts for the distri- 
bution of the lines of force. 

In Plate VIII. p. 488, fig. 1, let the vertical circle EE represent an 
electric current flowing from copper C to zinc Z through the conductor EE', 
as shewn by the arrows. 



472 ON PHYSICAL LINES OF FORCE. 

Let the homontal circle MM' represent a line of magnetic force embracing 
the electric circuit, the north and south directions being indicated by the lines 
SN and NS. 

Let the vertical circles V and V represent the molecular vortices of which 
the line of magnetic force is the axis. V revolves as the hands of a watch, 
and F' the opposite way. 

It will appear from this diagram, that if V and V were contiguous vortices, 
particles placed between them would move downwards ; and that if the particles 
were forced downwards by any cause, they would make the vortices revolve as 
in the figure. We have thus obtained a point of view from which we may 
regard the relation of an electric current to its lines of force as analogous to 
the relation of a toothed wheel or rack to wheels which it drives. 

In the first part of the paper we investigated the relations of the statical 
forces of the system. We have now considered the connexion of the motions 
of the parts considered as a system of mechanism. It remains that we should 
investigate the dynamics of the system, and determine the forces necessary to 
produce given changes in the motions of the different parts. 

Prop. VI. — To determine the actual energy of a portion of a medium due 
to the motion of the vortices within it. 

Let a, /8, y be the components of the circumferential velocity, as in Prop. II., 
then the actual energy of the vortices in unit of volume will be proportional 
to the density and to the square of the velocity. As we do not know the 
distribution of density and velocity in each vortex, we cannot determine the 
numerical value of the energy directly; but since /x also bears a constant 
though unknown ratio to the mean density, let us assume that the energy 
in unit of volume is 

where (7 is a constant to be determined. 
Let us take the case in which 

«=g. ^=f- y=t (^^)- 

Let <l> = <f>i + <f> (36), 



ON PHYSICAL LINES OF FORCE. 473 

then <^i is the potential at any point due to the magnetic system m„ and <^„ 
that due to the distribution of magnetism represented by m^. The actual 
energy of all the vortices is 

/; = 2C/x(a' + /8' + y)dF (38), 

the integration being performed over all space. 

This may be shewn by integration by parts (see Green's * Essay on Elec- 
tricity,' p. 10) to be equal to 

E= -4:iTCt{cf>,m,-h(f>,'m, + <f>,m, + (j>,m,)dV (39). 

Or since it has been proved (Green's 'Essay/ p. 10) that 

t<l>,m,dV=t<f>^m,dV, 

E=^-4:7rC{(fy{m, + <j),vi, + 2<f),m,)dV (40). 

Now let the magnetic system m^ remain at rest, and let w, be moved 
parallel to itself in the direction of x through a space Sx; then, since ^i 
depends on m^ only, it will remain as before, so that ^iTti^ will be constant ; 
and since <f>j depends on m, only, the distribution of (j), about m^ will remain 
the same, so that ^^rrij will be the same as before the change. The only part 
of E that will be altered is that depending on 2^,171^, because <^i becomes 

<^j 4- -p^ Zx on account of the displacement. The variation of actual energy due 
ux 

to the displacement is therefore 

hE=-inCt (2'^w,) dnx (41). 

But by equation (12) the work done by the mechanical forces on m^ during 
the motion is 



hW=t ("^^^dv) Bx (42); 



and since our hypothesis is a purely mechanical one, we must have by the 
conservation of force, 

hE+8W=0 (43); 

that is, the loss of energy of the vortices must be made up by work done iu 
moving magnets, so that 



AnCt (2 ^ m,dv\ Bx + X ("^ m,d v) Sx = 0, 



<^=l (^^)^ 

VOL. L 60 



474 ON PHYSICAL LINES OF FORCE. 

SO that the energy of the vortices in unit of volume is 

^/.(a' + ^ + y) (45); 

and that of a vortex whose volume is F is 

^^(a^ + /3^ + /)F. (46). 

In order to produce or destroy this energy, work must be expended on, 
or received from, the vortex, either by the tangential action of the layer of 
particles in contact with it, or by change of form in the vortex. We shall first 
investigate the tangential action between the vortices and the layer of particles 
in contact with them. 

Prop. VII. — To find the energy spent upon a vortex in unit of time by 
the layer of particles which surrounds it. 

Let P, Q, R be the forces acting on unity of the particles in the three 
co-ordinate directions, these quantities being functions of a;, y, and z. Since 
each particle touches two vortices at the extremities of a diameter, the reaction 
of the particle on the vortices will be equally divided, and will be 

-iP, -IQ, -iR 

on each vortex for unity of the particles; but since the superficial density of 
the particles is — (see equation (34)), the forces on unit of surface of a vortex 
will be 

"■4^^' "4^^' "4^^- 
Now let dS be an element of the surface of a vortex. Let the direction-cosines 
of the normal be I, m, n. Let the co-ordinates of the element be x, y, z. Let 
the component velocities of the surface be u, v, w. Then the work expended on 
that element of surface will be 

'^=-±(Fu + Qv + Rw)dS (47). 

Let us begin with the first term, PudS. P may be written 

^dP dP 

^^^d^'^^d^y 

and u^n^'-my. 



J, ^dP ^dP dP .... 

^" + ^^ + ^2/ + ^^ (48), 



ON PHYSICAL LINES OF FORCE. 



475 



Remembering that the surface of the vortex is a closed one, so that 

XnxdS = XmxdS = %mydS = tmzdS = 0, 
and XmydS = tnzdS= F, 

we find 2P^S=(f^-^r)F 

and the whole work done on the vortex in unit of time will be 
dE 1 



(49). 



^=-iz^(Pu + Qv + Rw)dS 

0.1 



An 



1 f /dQ dRX^^fdR dP\^ (dP dQ\\y 

:^Hd^-Wy)^^[dx-^z)^y[d^-dx)j^ 



47r l*Uz 



(50). 



Prop. VIII. — To find the relations between the alterations of motion of the 
vortices, and the forces P, Q, R which they exert on the layer of particles 
between them. 



Let V be the volume of a vortex, then by (46) its energy is 

1 



OTT 



and 



dE 1 Tr/ ^* . /o^/3_L ^y 



(51), 
.(52). 



(53). 



dt ' ^ dt ^ dtj 
Comparing this value with that given in equation (50), we find 

/dQ dR da\ , ^ /dR dP d^\ ^ fdP dQ dy\ . 

This equation being true for all values of a, ^, and y, first let yS and y 
vanish, and divide by a. We find 

dQ_dR_ da^ 
dz dy~^ dt 

^. ., , dR dP d^ 

and dP_dQ^ dry 

dy dx ^ dt 

From these equations we may determine the relation between the alterations 
of motion -j- , &c. and the forces exerted on the layers of particles between 

60—2 



(54). 



476 ON PHYSICAL LINES OF FORGE. 

the vortices, or, in the language of our hypothesis, the relation between changes 
in the state of the magnetic field and the electromotive forces thereby brought 
into play. 

In a memoir "On the Djoiamical Theory of Diffraction" (Cambridge Philo- 
sophical Transactions, Vol. ix. Part 1, section 6), Professor Stokes has given a 
method by which we may solve equations (54), and find P, Qy and R in tenns 
of the quantities on the right hand of those equations. I have pointed out* 
the application of this method to questions in electricity and magnetism. 

Let us then find three quantities F, G, H from the equations 



dG 

dz ~ 


dH "1 
■ dy =^'' 


dH 
dx 


dF ^ 


dF 
dy- 


dG 



(55), 



with the conditions '^\Ai^°'^'dy^^^dz^'^)^'^^^ ^^^^' 

dF dG dH ^ ,^^. 

dx dy dz 

Differentiating (55) with respect to t, and comparing with (54), we find 

-f . ^=f . -f (-)• 

We have thus determined three quantities, F, G, H, from which we can 
find P, Q, and R by considering these latter quantities as the rates at which 
the former ones vary. In the paper already referred to, I have given reasons 
for considering the quantities F, G, H as the resolved parts of that which 
Faraday has conjectured to exist, and has called the electrotonic state. In that 
paper I have stated the mathematical relations between this electrotonic state 
and the lines of magnetic force as expressed in equations (55), and also between 
the electrotonic state and electromotive force as expressed in equations (58). We 
must now endeavour to interpret them from a mechanical point of view in 
connexion with our hypothesis. 

* Camhridge Philosophical Transactions, Vol. X. Part i. Art. 3. "On Faraday's Lines of Force,' 
pp. 205—209 of this vol. 



ON PHYSICAL LINES OF FORCE. 477 

We shall in the first place examine the process by which the lines of force 
are produced by an electric current. 

Let AB, Plate VIII. , p. 488, fig. 2, represent a current of electricity in the 
direction from A to B. Let the large spaces above and below AB represent the 
vortices, and let the small circles separating the vortices represent the layers of 
particles placed between them, which in our hypothesis represent electricity. 

Now let an electric current from left to right commence in AB. The 
row of vortices gh above AB will be set in motion in the opposite direction 
to that of a watch. (We shall call this direction +, and that of a watch -.) 
We shall suppose the row of vortices kl still at rest, then the layer of particles 
between these rows will be acted on by the row gh on their lower sides, and 
will be at rest above. If they are free to move, they will rotate in the 
negative direction, and will at the same time move from right to left, or in 
the opposite direction from the current, and do form an induced electric current. 

If this current is checked by the electrical resistance of the medium, the 
rotating particles will act upon the row of vortices Jcl, and make them revolve 
in the positive direction till they arrive at such a velocity that the motion of 
the particles is reduced to that of rotation, and the induce4 current disappears. 
If, now, the primary current AB be stopped, the vortices in the row gh will 
be checked, while those of the row kl still continue in rapid motion. The 
momentum of the vortices beyond the layer of particles pq will tend to move 
them from left to right, that is, in the direction of the primary current; but 
if this motion is resisted by the medium, the motion of the vortices beyond pq 
will be gradually destroyed. 

It appears therefore that the phenomena of induced currents are part of the 
process of communicating the rotatory velocity of the vortices from one part of 
the field to another. 

As an example of the action of the vortices in producing induced currents, 
let us take the following case :— Let B, Plate VIIL, p. 488, fig. 3, be a circular 
ring, of uniform section, lapped uniformly with covered wire. It may be shewn 
that if an electric current is passed through this wire, a magnet placed within 
the coil of wire wiU be strongly affected, but no magnetic effect wUl be produced 
on any external point. The effect will be that of a magnet bent round till 
its two poles are in contact. 

If the coil is properly made, no effect on a magnet placed outside it can 



478 ON PHYSICAL LINES OF FORCE. 

be discovered, whether the current is kept constant or made to vary in strength ; 
but if a conducting wire C be made to embrace the ring any number of times, 
an electromotive force will act on that wire whenever the current in the coil is 
made to vary ; and if the circuit be closed^ there will be an actual current in 
the wire C. 

This experiment shews that, in order to produce the electromotive force, it 
is not necessary that the conducting wire should be placed in a field of magnetic 
force, or that lines of magnetic force should pass through the substance of the 
wu'e or near it. All that is required is that lines of force should pass through 
the circuit of the conductor, and that these lines of force should vary in quantity 
during the experiment. 

In this case the vortices, of which we suppose the lines of magnetic force 
to consist, are all within the hollow of the ring, and outside the ring all is at 
rest. If there is no conducting circuit embracing the ring, then, when the 
primary current is made or broken, there is no action outside the ring, except 
an instantaneous pressure between the particles and the vortices which they 
separate. If there is a continuous conducting circuit embracing the ring, then, 
when the primary current is made, there will be a current in the opposite 
direction through C; and when it is broken, there will be a current through C 
in the same direction as the primary current. 

We may now perceive that induced currents are produced when the elec- 
tricity yields to the electromotive force, — this force, however, still existing 
when the formation of a sensible current is prevented by the resistance of the 
circuit. 

The electromotive force, of which the components are P, Q, R, arises from 
the action between the vortices and the interposed particles, when the velocity 
of rotation is altered in any part of the field. It corresponds to the pressure 
on the axle of a wheel in a machine when the velocity of the driving wheel 
is increased or diminished. 

The electrotonic state, whose components are F, G, H, is what the electromotive 
force would be if the currents, &c. to which the lines of force are due, instead 
of arriving at their actual state by degrees, had started instantaneously from 
rest with their actual values. It corresponds to the impulse which would act 
on the axle of a wheel in a machine if the actual velocity were suddenly given 
to the driving wheel, the machine being previously at rest. 



ON PHYSICAL LINES OF FORCE. 479 

If the machine were suddenly stopped by stopping the driving wheel, each 
wheel would receive an impulse equal and opposite to that which it received 
when the machine was set in motion. 

This impulse may be calculated for any part of a system of mechanism, 
and may be called the reduced momentum of the machine for that point. In 
the varied motion of the machine, the actual force on any part arising from 
the variation of motion may be found by diiferentiating the reduced momentum 
with respect to the time, just as we have found that the electromotive force 
may be deduced from the electrotonic state by the same process. 

Having found the relation between the velocities of the vortices and the 
electromotive forces when the centres of the vortices are at rest, we must 
extend our theory to the case of a fluid medium containing vortices, and 
subject to all the varieties of fluid motion. If we fix our attention on any 
one elementary portion of a fluid, we shall find that it not only travels from 
one place to another, but also changes its form and position, so as to be elon- 
gated in certain directions and compressed in others, and at the same time (in 
the most general case) turned round by a displacement of rotation. 

These changes of form and position produce changes in the velocity of the 
molecular vortices, which we must now examine. 

The alteration of form and position may always be reduced to three simple 
extensions or compressions in the direction of three rectangular axes, together 
with three angular rotations about any set of three axes. We shall first con- 
sider the effect of three simple extensions or compressions. 

Prop. IX. — To find the variations of a, yS, y in the parallelepiped .r, y, z 
when X becomes x-^-hx; y, y + Sy ; and z, z + Bz; the volume of the figure 
remaining the same. 

By Prop. II. we find for the work done by the vortices against pressure, 

hW=p,B{xyz)-^(a'yzBx-i-p:'zxZy-\-'/x2jSz) (59); 

and by Prop. VI. we find for the variation of energy, 

BE=-^(aBa + ^h^-{-yBy)xyz (60). 

477 



480 ON PHYSICAL LINES OF FORCE. 

The sum SW+BE must be zero by the conservation of energy, and 8 (xyz) = 0, 
since xyz is constant; so that 



(Sa-af)+^(s^-^|)+y(Sy-y|) = (61). 



In order that this should be true independently of any relations between a, /8, 
and y, we must have 

Sa = a«|, S^=;8j, Sy = y| (62). 

Prop. X. — To find the variations of a, /8, y due to a rotation 0^ about the 
axis of X from y to 2;, a rotation O^ about the axis of y from z to x, and a 
rotation ^3 about the axis of z from ic to y. 

The axis of y8 will move away from the axis of x by an angle $3 ; so 
that /8 resolved in the direction of x changes from to —JSO^. 

The axis of y approaches that of x by an angle 6^ ; so that the resolved 
part of y in direction x changes from to yd^. 

The resolved part of a in the direction of x changes by a quantity depending 
on the second power of the rotations, which may be neglected. The variations of 
a, )8, y from this cause are therefore 

8a = yl9,-M, S^ = a^3-y(9„ hy^^d.-aO, (63). 

The most general expressions for the distortion of an element produced by 
the displacement of its different parts depend on the nine quantities 

d ^ d ^ d ^ d ^ d ^ d ^ d ^ d ^ d ^ 

tJ""' 3^^^' Tz^"' Tx^J- Ty^y- di^' Tx^' Ty^- Tz^-' 

and these may always be expressed in terms of nine other quantities, namely, 
three simple extensions or compressions, 

Zx Zy hz' 
^' Y' ~^ 
along three axes properly chosen, x\ y\ z', the nine direction-cosines of these 
axes with their six connecting equations, which are equivalent to three inde- 
pendent quantities, and the three rotations 6^, 0,, 0^ about the axes of x, y, z. 

Let the direction-cosines of x' with respect to cc, y, z be /„ mj, n^^ those of 
y\ \y 7?ij, Tiy and those of z\ Zj, ma, n, ; then we find 



ON PHYSICAL LINES OF FORCE. 481 



dx X y z 

-J- Bx = I,m, — + km, 4- + ^wi,— - d. 



dy ' ' X ' ' y 



(C4), 



witli similar equations for quantities involving Sy and 8z. 

Let a, 13', y be the values of a, ^, y referred to the axes x, y, z; then 

a=l,a + mJ3 + n,y^ 

^' = l,a + mS-^n,y I (65). 

y = l,a + m^fi + n{y J 
We shaU then have ha = kha +a^ ^-l,^' + ye,-^e, {^(:>), 

=i^a'^+ij3'K+W^f+ye.-^d. (67). 

By substituting the values of a, /3', y, and comparing with equations (64), we 
find 

^- = 4^--^4'"^^^'^ ^''^ 

as the variation of a due to the change of form and position of the element. 
The variations of ^ and y have similar expressions. 

Prop. XI.— To find the electromotive forces in a moving body. 

The variation of the velocity of the vortices in a moving element is due to 
two causes— the action of the electromotive forces, and the change of form and 
position of the element. The whole variation of a is therefore 

«"=KS-f)^'^"^^^^^4^^^^^^^ ^''\ 

But since a is a function of x, y. z and t, the variation of a may be aiso written 

^'^=Pj^-py-^TJ'^'i^ (^»'- 

Equating the two values of Sa and dividing by ht, and remembering that in the 

motion of an incompressible medium 

d dx ddy d dz_ /^,x 

didt^dy dt^ dzdt~^ ^ ^' 

vol. l ^1 



482 



ON PHYSICAL LINES OF FORCE. 



id that in the absence of free magnetism 



dx dy dz 



•(72). 



we find 
\/dQ 
fi\dz 



Putting 



dy) 



d dx 
'^'^dz'dt 



d dz d dy ^n d dx 
'^dzdt~°'dydi ' '^'dyTt 



dy dx da dz da dy d^ dx _^da _ 
dz dt dz dt dy dt dy dt di 



and 





l/dG 

^'ATz- 

_\(d'G 
~ fi \dz dt 


dH\ 


da 
di- 


dy)" 

d'H\ 
dydt) 



,.(r.3). 

• (74), 
..(75). 



where F, G, and H are the vahies of the electrotonic components for a fixed 
point of space, our equation becomes 



dx 



dz dG 



Q + l^y:J7-H-^7u- 



d fry dy r^dx 



f) = o (-)• 



dy V^" ' '^^ dt '^'^ dt 

The expressions for the variations of ^ and y give us two other equations 
which may be written down from symmetry. The complete solution of the three 
equations is 



Q 



dz 
dt 

dx 



dF _d^ 
dt dx 

^. dG _d^ 
di ^'^ dt "^ dt dy 

dll_d^ 
dz 



„ ^dx dy 



(77). 



The first and second terms of each equation indicate the effect of the motion 
of any body in the magnetic field, the third term refera to changes in the 
electrotonic state produced by alterations of position or intensity of magnets 
or currents in the field, and ^ is a function of x, y, z, and t, which is inde- 
terminate as far as regards the solution of the original equations, but which 
may always be determined in any given case from the circumstances of the 
problem. The physical interpretation of ^ is, that it is the clectiic tension at 
each point of space. 



ON PHYSICAL LINP:S OF FORCE. 483 

The physical meaning of the terms in the expression for the electromotive 
force depending on the motion of the body, may be made simpler by supposing 
the field of magnetic force uniformly magnetized with intensity a in the direction 
of the axis of x. Then if /, m, n be the direction-cosines of any portion of a 
linear conductor, and S its length, the electromotive force resolved in the direction 
of the conductor will be 

e = S{Pl + Qm + Rn) (78), 

' = ^^^{'''jt-''t) (^^)' 

that is, the product of /xa, the quantity of magnetic induction over unit of area 

multiplied by Sim y, -" ;7r)» the area swept out by the conductor S in unit of 

time, resolved perpendicular to the direction of the magnetic force. 

The electromotive force in any part of a conductor due to its motion is 
therefore measured by the number of lines of magnetic force which it crosses 
in unit of time ; and the total electromotive force in a closed conductor is 
measured by the change of the number of lines of force which pass through it ; 
and this is true whether the change be produced by the motion of the con- 
ductor or by any external cause. 

In order to understand the mechanism by which the motion of a conductor 
across lines of magnetic force generates an electromotive force in that conductor, 
we must remember that in Prop. X. we have proved that the change of form 
of a portion of the medium containing vortices produces a change of the velocity 
of those vortices ; and in particular that an extension of the medium in the 
direction of the axes of the vortices, combined with a contraction in all direc- 
tions perpendicular to this, produces an increase of velocity of the vortices ; 
while a shortening of the axis and bulging of the sides produces a diminution 
of the velocity of the vortices. 

This change of the velocity of the vortices arises from the internal effects 
of change of form, and is independent of that produced by external electro- 
motive forces. If, therefore, the change of velocity be prevented or checked, 
electromotive forces will arise, because each vortex will press on the surrounding 
particles in the direction in which it tends to alter its motion. 

Let A, fig. 4, p. 488, represent the section of a vertical wire moving in the 
direction of the arrow from west to east, across a system of lines of magnetic force 

61—2 



484 ON PHYSICAL LINES OF FORCE. 

running north and south. The curved lines in fig. 4 represent the lines of fluid 
motion about the wire, the wire being regarded as stationary, and the fluid as 
having a motion relative to it. It is evident that, from this figure, we can trace 
the variations of form of an element of the fluid, as the form of the element 
depends, not on the absolute motion of the whole system, but on the relative 
motion of its parts. 

In front of the wire, that is, on its east side, it will be seen that as the 
wire approaches each portion of the medium, that portion is more and more 
compressed in the direction from east to west, and extended in the direction 
from north to south ; and since the axes of the vortices lie in the north and 
south direction, their velocity will continually tend to increase by Prop. X., 
unless prevented or checked by electromotive forces acting on the circumference 
of each vortex. 

We shall consider an electromotive force as positive when the vortices tend 
to move the interjacent particles upwards perpendicularly to the plane of the 
paper. 

The vortices appear to revolve as the hands of a watch when we look at 
them from south to north ; so that each vortex moves upwards on its west side, 
and downwards on its east side. In front of the wire, therefore, where each 
vortex is striving to increase its velocity, the electromotive force upwards must 
be greater on its west than on Its east side. There will therefore be a con- 
tinual increase of upward electromotive force from the remote east, where it is 
zero, to the front of the moving wire, where the upward force wiU be strongest. 

Behind the wire a difierent action takes place. As the wire moves away 
from each successive portion of the medium, that portion is extended from east 
to west, and compressed from north to south, so as to tend to diminish the 
velocity of the vortices, and therefore to make the upward electromotive force 
greater on the east than on the west side of each vortex. The upward electro- 
motive force wiU therefore increase continually from the remote west, where it 
is zero, to the back of the moving wire, where it will be strongest. 

It appears, therefore, that a vertical wire moving eastwards will experience 
an electromotive force tending to produce in it an upward current. If there 
is no conducting circuit in connexion with the ends of the wire, no current will 
be formed, and the magnetic forces wHl not be altered ; but if such a circuit 
exists, there will be a current, and the lines of magnetic force and the velocity 



ON PHYSICAL LINES OF FORCE. 485 

of the vortices will be altered from their state previous to the motion of the 
wire. The change in the lines of force is shewn in fig. 5. The vortices in 
front of the wire, instead of merely producing pressures, actually increase in 
velocity, while those behind have their velocity diminished, and those at the 
sides of the wire have the direction of their axes altered; so that the final 
effect is to produce a force acting on the wire as a resistance to its motion. 
We may now recapitulate the assumptions we have made, and the results we 
have obtained. 

(1) Magneto-electric phenomena are due to the existence of matter under 
certain conditions of motion or of pressure in every part of the magnetic field, 
and not to direct action at a distance between the magnets or currents. The 
substance producing these effects may be a certain part of ordinary matter, or 
it may be an aether associated with matter. Its density is greatest in iron, 
and least in diaraagnetic substances ; but it must be in all cases, except that of 
iron, very rare, since no other substance has a large ratio of magnetic capacity 
to what we call a vacuum. 

(2) The condition of any part of the field, through which lines of magnetic 
force pass, is one of unequal pressure in different directions, the direction of 
the lines of force being that of least pressure, so that the lines of force may 
be considered lines of tension. 

(3) This inequality of pressure is produced by the existence in the medium 
of vortices or eddies, having their axes in the direction of the lines of force, 
and having their direction of rotation determined by that of the lines of force. 

We have supposed that the direction was that of a watch to a spectator 
looking from south to north. We might with equal propriety have chosen the 
reverse direction, as far as known facts are concerned, by supposing resinous elec- 
tricity instead of vitreous to be positive. The effect of these vortices depends 
on their density, and on their velocity at the circumference, and is independent 
of their diameter. The density must be proportional to the capacity of the 
substance for magnetic induction, that of the vortices in air being 1. The 
velocity must be very great, in order to produce so powerful effects in so rare 
a medium. 

The size of the vortices is indeterminate, but is probably very small as 
compared with that of a complete molecule of ordinary matter^''. 

* The angular momentum of the system of vortices depends on their average diameter ; so tkat if the 
diameter were sensible, we might expect that a magnet would behave as if it contained a revoh-ing bodv 



486 O^' PHYSICAL LINES OF FORCE. 

(4) The vortices are separated from each other by a single layer of round 
particles, so that a system of cells is formed, the partitions being these layers 
of particles, and the substance of each cell being capable of rotating as a vortex. 

(5) The particles forming the layer are in rolling contact with both the 
vortices which they separate, but do not rub against each other. They are 
perfectly free to roll between the vortices and so to change their place, provided 
they teep within one complete molecule of the substance; but in passing from 
one molecule to another they experience resistance, and generate irregular 
motions, which constitute heat. These particles, in our theory, play the part of 
electricity. Their motion of translation constitutes an electric current, their 
rotation serves to transmit the motion of the vortices from one part of the 
field to another, and the tangential pressures thus called into play constitute 
electromotive force. The conception of a particle having its motion connected 
with that of a vortex by perfect rolling contact may appear somewhat awkward. 
I do not bring it forward as a mode of connexion existing in nature, or even 
as that which I would willingly assent to as an electrical hypothesis. It is, 
however, a mode of connexion which is mechanically conceivable, and easily 
investigated, and it serves to bring out the actual mechanical connexions 
between the known electro-magnetic phenomena; so that I venture to say that 
any one who understands the provisional and temporary character of this 
hypothesis, will find himself rather helped than hindered by it in his search 
after the true interpretation of the phenomena. 

The action between the vortices and the layers of particles is in part 
tangential; so that if there were any slipping or difierential motion between 
the parts in contact, there would be a loss of the energy belonging to the 
lines of force, and a gradual transformation of that energy into heat. Now we 
know that the hues of force about a magnet are maintained for an indefinite 
time without any expenditure of energy; so that we must conclude that 
wherever there is tangential action between difierent parts of the medium, there 
is no motion of slipping between those parts. We must therefore conceive that 
the vortices and particles roll together without shpping ; and that the interior 
strata of each vortex receive their proper velocities from the exterior stratum 
without slipping, that is, the angular velocity must be the same throughout each 
vortex. 

within it, -and that the existence of this rotation might be detected by experiments on the free rotation of a 
magnet. I have made experiments to investigate this question, but have not yet fully tried the apparatus. 



ON PHYSICAL LINES OF FORCE. 487 

The only process in which electro- magnetic energy is lost and transforaied 
into heat, is in the passage of electricity from one molecule to another. In all 
other cases the energy of the vortices can only be diminished when an equivalent 
quantity of mechanical work is done by magnetic action. 

(6) The effect of an electric current upon the surrounding medium is to 
make the vortices in contact with the current revolve so that the parts next 
to the current move in the same direction as the current. The parts furthest 
from the current will move in the opposite direction ; and if the medium is a 
conductor of electricity, so that the particles are free to move in any direction, 
thfe particles touching the outside of these vortices will be moved in a direction 
contrary to that of the current, so that there will be an induced current in 
the opposite direction to the primary one. 

If there were no resistance to the motion of the particles, the induced 
current would be equal and opposite to the primary one, and would continue 
as long as the primary current lasted, so that it would prevent all action of 
the primary current at a distance. If there is a resistance to the induced 
current, its particles act upon the vortices beyond them, and transmit the motion 
of rotation to them, till at last all the vortices in the medium are set in 
motion with such velocities of rotation that the particles between them have no 
motion except that of rotation, and do not produce currents. 

In the transmission of the motion from one vortex to another, there arises u 
force between the particles and the vortices, by which the particles are pressed 
in one direction and the vortices in the opposite direction. We call the force 
actino- on the particles the electromotive force. The reaction on the vortices is 
equal and opposite, so that the electromotive force cannot move any part of 
the medium as a whole, it can only produce currents. When the primary 
current is stopped, the electromotive forces all act in the opposite direction. 

(7) When an electric current or a magnet is moved in presence of a 
conductor, the velocity of rotation of the vortices in any part of the field is 
altered by that motion. The force by which the proper amount of rotation is 
transmitted to each vortex, constitutes in this case also an electromotive force, 
and, if permitted, will produce currents. 

(8) When a conductor is moved in a field of magnetic force, the vortices 
in it and in its neighbourhood are moved out of their places, and are changed 
in form. The force arising from these changes constitutes the electromotive 



488 ON PHYSICAL LINES OF FORCE. 

force on a moving conductor, and is found by calculation to correspond with 
that determined by experiment. 

"We have now shewn in w-hat way electro -magnetic phenomena may be 
imitated by an imaginary system of molecular vortices. Those who have been 
already inclined to adopt an hypothesis of this kind, will find here the con- 
ditions which must be fulfilled in order to give it mathematical coherence, and 
a comparison, so far satisfactory, between its necessary results and known facts. 
Those who look in a different direction for the explanation of the facts, may 
be able to compare this theory with that of the existence of currents flowing 
freely through bodies, and with that which supposes electricity to act at a 
distance with a force depending on its velocity, and therefore not subject to 
the law of conservation of energy. 

The facts of electro-magnetism are so complicated and various, that the 
explanation of any number of them by several different hypotheses must be 
interesting, not only to physicists, but to all who desire to understand how 
much evidence the explanation of phenomena lends to the credibility of a theory, 
or how far we ought to regard a coincidence in the mathematical expression of 
two sets of phenomena as an indication that these phenomena are of the same 
kind. We know that partial coincidences of this kind have been discovered ; 
and the fact that they are only partial is proved by the divergence of the 
laws of the two sets of phenomena in other respects. We may chance to find, 
in the higher parts of physics, instances of more complete coincidence, which 
may require much investigation to detect their ultimate divergence. 



NOTE. 

Since the first part of this paper was written, I have seen in Crelle's Journal for 1859, 
a paper by Prof. Helmholtz on Fluid Motion, in which he has pointed out that the lines 
of fluid motion are arranged according to the game laws as the Hnes of magnetic force, the 
path of an electric current corresponding to a line of axes of those particles of the fluid 
which are in a state of rotation. This is an additional instance of a physical analogy, the 
investigation of which may illustrate both electro-magnetism and hydrodynamics. 




^ ^ 






Fig 5 



VOL. L PLATE VIIL 



JFi^: 7. 



Ti^ S. 



XigJD. 




Tvg:9. a. 



ON PHYSICAL LINES OF FORCE. 489 



[From the Philosophical Magazine for January and February, 1802.] 
PART III. 

THE THEORY OF MOLECULAR VORTICES APPLIED TO STATICAL ELECTRICITY. 

In the first part of this paper ^^ I have shewn how the forces acting between 
ma^ets, electric currents, and matter capable of magnetic induction may be 
accounted for on the hypothesis of the magnetic field being occupied with 
innumerable vortices of revolving matter, their axes coinciding with the direction 
of the magnetic force at every point of the field. 

The centrifugal force of these vortices produces pressures distributed in such 
a way that the final efiect is a force identical in direction and magnitude 
with that li^ich we observe. 

In the second partf I described the mechanism by which these rotations 
may be made to coexist, and to be distributed according to the known laws 
of magnetic lines of force. 

I conceived the rotating matter to be the substance of certain cells, divided 
from each other by cell-walls composed of particles which are very small com- 
pared with the cells, and that it is by the motions of these particles, and their 
tangential action on the substance in the cells, that the rotation is communi- 
cated from one cell to another. 

I have not attempted to explain this tangential action, but it is necessary 

to suppose, in order to account for the transmission of rotation from the exterior 

to the interior parts of each cell, that the substance in the cells possesses 

elasticity of figure, similar in kind, though different in degree, to that observed 

in BoUd bodies. The undulatory theory of light requires us to admit this kind 

of elasticity in the luminiferous medium, in order to account for transverse 

vibrations. We need not then be surprised if the magneto-electric medium 

possesses the same property. 

♦ PhiL Mag. March, 1861 [pp. 4.51— 466 of this vol.]. 

t Phil. Mag. April and May, 1861 [pp. 467—488 of this vol.]. 

VOL. I. ^- 



490 ON PHYSICAL LINES OF FORCE. 

According to our theory, the particles which forta the pai-titions between 
the cells constitute the matter of electricity. The motion of these particles 
constitutes an electric current; the tangential force with which the particles 
are pressed by the matter of the cells is electromotive force, and the pressure 
of the particles on each other corresponds to the tension or potential of the 
electricity. 

If we can now explain the condition of a body with respect to the 
surrounding medium when it is said to be "charged" with electricity, and 
account for the forces acting between electrified bodies, we shall have established 
a connexion between all the principal phenomena of electrical science. 

We know by experiment that electric tension is the same thing, whether 
observed in statical or in current electricity; so that an electromotive force 
produced by magnetism may be made to charge a Leyden jar, as Ls done by 
the coil machine. 

When a difference of tension exists in different parts of any body, the 
electricity passes, or tends to pass, from places of greater to places of smaller 
tension. If the body is a conductor, an actual passage of electricity takes 
place; and if the difference of tensions is kept up, the current continues to 
flow with a velocity proportional inversely to the resistance, or directly to the 
conductivity of the body. 

The electric resistance has a very wide range of values, that of the metals 
being the smallest, and that of glass being so great that a charge of electricity 
has been preserved'"* in a glass vessel for years without penetrating the thick- 
ness of the glass. 

Bodies which do not permit a current of electricity to flow through them 
are called insulators. But though electricity does not flow through them, 
the electrical effects are propagated through them, and the amount of these 
effects differs according to the nature of the body; so that equally good insu- 
lators ma